Sheldon  &  Company's   Text~:Sooks, 


PROFESSOR    OLNEY'S 


NEW   MATHEMATICAL    SERIES 


The  success  of  Prof.  Olney's  series  has  been  most  wonderful. 

With  all  their  admitted  excellencies,  both  the  Author  and  Pub- 
lishers have  felt  that  it  was  possible  to  retain  their  many  attractive 
features  and  yet  adapt  the  books  more  perfectly  to  the  special 
school-room  wants. 

To  accomplish  this  most  desirable  end,  Professor  Olney  has  been 
accumulating  very  valuable  suggestions.  He  has  also,  for  several 
years,  had  associated  with  him  in  the  preparation  of  this  new 
series,  some  of  the  best  practical  teachers  in  the  country. 

The  design  is  to  present  to  the  educational  public  the  best  and 
most  teachable  series  of  Mathematics  ever  published.  The  work 
is  now  so  far  advanced  that  the  Publishers  are  able  to  make  the 
above  pleasing  announcement,  which  they  feel  will  be  of  great 
interest  to  all  who  are  engaged  in  teaching. 


THE    NEW  SERIES    EMBRACES: 

I. 

Olney' s  First  Lessons  in  Arithfuetic.    Just  Published. 

II. 
Olriey's  Practical  Arithinetic. 

This  book  has  been  published  but  a  short  time,  but  it  has 
already  had  the  most  wonderful  success. 
They  are  models  of  beauty  and  cheapness. 


For  schools  of  a  high  grade,  Professor  Olney  has  prepared — 

III. 
The  Science  of  Arithmetic. 

lY. 

The  First  Frincij^les  of  Algebra. 

An  Introduction  to   the    Author's  Complete  and 
University  Algebras. 

V. 

Olney' s  Complete  Algebra,    New  Edition,  in  large  type. 

This  book  is  now  entirely  re-el ectrotyped  in  larger  and  more 

attractive  type.    Theexplanatory  matter  is  greatly  lessened.    The 

attractive  features  of  this  book,  which  have  made  it  the  most 

popular  Algebra  ever  published  in  this  country,  are  all  retained. 


.051 


Sheldon  &    Company's   Text-^Books^ 


OLNEY'S  SERIES  OF  MATHEMATICS. 

Olney's  First  Lessons  in  Arithinetlc  Illus- 
trated  

Olney's  Practical  Arithmetic 

Olney's  Science  of  Arithmetic,  (For  High-Schools 
only.) 

Olney^s  First  Principles  of  Algebra 

Olney's  Complete  Algebra 

Olney's  Book  of  Test  Examples  in  Algebra.., 

Olney's  University  Algebra 

Olney's  Elements  Geom,  4&  Trigonom.  (Sch.  Ed.) 

Olney's  Elements  of  Geometry.    Separate 

Olney's  Elements  of  Trigonometry.    Separate. . 

Olney's  Elements  of  Geometry  and  Trigonom," 
etry.    (Univ.  Ed.,  with  Tables  of  Logarithms.) 

Olney's  Elements  of  Geometry  and  Trigonom- 
etry.   (University  Edition,  without  Tables.) 

Olney's  General  Geometry  and  Calculus 

The  universal  favor  with  which  these  books  have  been  received 
by  educators  in  all  parts  of  the  country,  leads  the  publishers  to 
think  that  they  have  supplied  a  felt  want  in  our  educational  ap- 
pliances. 

There  is  one  feature  which  characterizes  this  series,  so  unique, 
and  yet  so  eminently  practical,  that  we  feel  desirous  of  calling 
special  attention  to  it.     It  is 

The  facility  with  which  the  books  can  be  used  for  classes  of 

all  grades,  and  in  schools  of  the  widest 

diversity  of  purpose. 

Each  volume  in  the  series  is  so  constructed  that  it  may  be  used 
with  equal  ease  by  the  youngest  and  least  disciplined  who  should 
fee  pursuing  its  theme,  and  by  those  who  in  more  mature  years 
and  with  more  ample  preparation  enter  upon  the  study. 


Ij 


Library  ciW.H.Metzleri 
Class  Noi 


OLNEY'S   MATHEMATICAL    SERIES. 

A  W.    I' 

GENERAL    GEOMETRY 

AND 

CALCULUS. 


INCIitTDING  BOOK  I.    OF  THE   GENEEAIj   GEOMETRY,    TREATING  OP  LOCI  IN 
A  PLANE  ;    AND   AN  ELEMENTARY   COURSE   IN  THE   DIFFER- 
ENTIAL AND  INTEGRAL   CALCULUS. 


BT 


EDWAED  OLNEY, 

FBOFESSOB  OF  MATHEMATICS  IN  THE  Xmi^^^^F^^jpOMOeUXj 


NEW  YOEK : 
SHELDON    AND     COMPANY. 

1881. 


Entered  according  to  Act  of  Congress  in  the  year  1871,  by 

SHELDON   &   COMPANY, 

In  the  Office  of  the  Librarian  of  Congress  at  Washington. 


PROF.  OLNEY'S  MATHEMATICAL  COURSE. 


INTRODUCTION  TO  ALGEBRA 

COMPLETE  ALGEBRA 

KEY  TO  COMPLETE  ALGEBRA  --,---. 

UNIVERSITY  ALGEBRA 

KEY  TO  UNIVERSITY  ALGEBRA 

A  VOLUME  OF  TEST  EXAMPLES  IN  ALGEBRA 

ELEMENTS  OF  GEOMETRY  AND  TRIGONOMETRY 

ELEMENTS  OF  GEOMETRY  AND  TRIGONOMETRY,  University 
Edition - 

ELEMENTS  OF  GEOMETRY,  separate    - 

ELEMENTS  OF  TRIGONOMETRY,  separate 

GENERAL  GEOMETRY  AND  CALCULUS    -       -       -       - 

BELLOWS'  TRIGONOMETRY 

PROF.  OLNEY'S  SERIES  OF  ARITHMETICS. 

PRIMARY  ARITHMETIC 

ELEMENTS  OP  ARITHMETIC 

PRACTICAL  ARITHMETIC 

SCIENCE  OF  ARITHMETIC       -        -        -        -       -        -       - 

G.  &.  C. 

150249 


PREFACE. 


-•♦i- 


^  This  volume  presents  a  course  in  the  General  Geometry  and 
the  Infinitesimal  Calculus,  which  is  thought  to  be  as  extended  as 
is  pr,acticable  for  the  general  student  in  the  regular  undergrad- 
uate course  in  our  American  colleges.  If  we  can  secure  a  suffi- 
ciently high  grade  of  preparation,  so  that  students  in  the  Fresh- 
man year  can  complete  a  respectable  course  in  Elementary 
Geometry,  including  Plane  and  Spherical  Trigonometry,  and  in 
Algebra,  it  is  thought  that  during  the  Sophomore  year  the  3on- 
tents  of  this  volume  can  be  readily  mastered.  Such  is  the 
purpose  in  this  University ;  and  it  is  already  well  nigh  reahzed. 

As  to  the  propriety  of  including  the  study  of  both  these  sub- 
jects in  the  regular  undergraduate  course,  there  can  be  but  one 
opinion  among  those  competent  to  judge.  No  man  can  justly 
claim  to  have  a  good  general  education,  who  is  ignorant  of  the 
elements  of  the  processes  by  which  all  extended  operations  in 
the  exact  sciences  are  carried  forward,  and  which  are  the  foun- 
dation of  all  the  arts  based  upon  mathematical  science.  The 
man  who  is  ignorant  of  the  General  Geometry  and  the  Calculus, 
is  not  only  a  stranger  in  one  of  the  subHmest  realms  of  human 
thought,  but  knows  nothing  of  the  instruments  in  most  familiar 
use  by  the  engiueer,  the  astronomer,  and  the  machinist  in  any  of 
the  higher  walks  of  art.  In  short  he  is  ignorant  of  the  charac- 
teristic processes  of  the  mathematician  of  his  day. 

Nor  is  it  impracticable  for  the  majority  of  students  to  become 
intelligent  in  these  subjects.  They  do  not  he  beyond  the  reach 
of  good  common  minds,  nor  require  peculiar  mental  character- 
istics for  their  mastery.  The  difficulty  hitherto  has  been  in  the 
methods  of  presentation,  in  the  limited  and  totally  inadequate 
amount  of  time  assigned  them,  and  more  than  all  in  the  precon- 
ceived notion  of  their  abstruseness. 

The  mathematician  will  see  in  the  plan  of  the  first  part  of 


IV  PREFACE, 

this  volume,  as  well  as  in  its  title  ( General  Geometry),  a  recog- 
nition of  the  profound  views  of  Comte  upon  the  philosophy  of 
the  science.  This  science  is  a  method  of  Geometrical  reasoning. 
Its  characteristic  feature  is  that  it  represents  form,  as  well  as 
magnitude,  by  equations,  and  hence  makes  algebra  its  instru- 
ment. It  is  consequently  indirect.  Its  ultimate  object  is  breadth 
of  comprehension, — the  discussion  of  general  problems.  In 
accordance  with  this  conception,  the  first  purpose  is  to  exhibit 
the  method  of  translating  geometrical  forms  into  algebraic 
equations,  i.  6.  to  show  how  loci  are  represented  by  equations. 
While  the  prominence  is  given  to  the  Conic  Sections  which 
their  importance  in  physical  science  demands,  the  student  is 
not  led  to  think  that  this  is  merely  a  scheme  for  treating  these 
curves.  He  is  taught  to  look  upon  it  as  a  method  of  investiga- 
tion— as  designed  to  embrace  the  discussion  of  all  loci.  For  this 
purpose  many  Higher  Plane  Curves  are  treated.  After  the 
student  has  become  famihar  with  the  equation  as  the  represent- 
ative of  a  locus,  and  has  learned  how  to  ]3roduce  the  equation  of 
a  locus  from  its  definition,  he  has  obtained  the  instrument.  He 
is  now  to  learn  how  to  apply  it  for  the  purposes  of  Geometrical 
investigation.  In  carrying  forward  this  part  of  his  study  the 
Calculus  renders  invaluable  service.  Moreover,  this  preparatory 
study  of  the  General  Geometry  gives  him  exactly  the  needed 
means  for  illustrating  the  elementary  processes  of  the  Calculus. 
He  has,  therefore,  come  to  a  point  where  his  further  progress 
requires  a  knowledge  of  the  Differential  Calculus,  and  he  has 
also  the  requisite  preparation  for  its  study.  Hence,  after  having 
become  famihar  with  the  first  three  chapters  of  General  Geom- 
etry, he  reads  the  Differential  Calculus. 

By  this  arrangement  the  Calculus  is  seen  in  its  true  relations, 
as  an  independent  abstract  science,  grand  and  beautiful  in  itself, 
and  rendering  most  efficient  service  in  the  more  immediately 
practical  science  of  Geometry,  as  it  is  afterwards  seen  to  do  in 
Physics.  Having  obtained  the  needed  acquaintance  with  the 
Differential  Calculus,  the  student  returns  to  pursue  his  Geomet- 
rical studies,  with  equations  of  loci  as  his  instruments,  and  the 
Calculus  to  aid  in  the  manipulation  of  them.  But  the  pecuhar 
features  of  the  treatise  are  too  numerous  to  be  enumerated  here, 
and  can  be  seen  in  their  true  light  only  by  a  perusal  of  the 
work. 


PBEFACE.  V 

In  the  treatment  of  the  Calculus  I  have  used  the  Infinitesi- 
mal method  instead  of  the  method  of  Limits,  on  account  of  its 
greater  simplicity,  as  well  as  because  it  is  the  only  conception 
which  enables  us  to  apply  the  Calculus  to  practical  problems 
with  any  degree  of  facility.  The  general  use  of  the  method  of 
limits  in  our  text  books  has  done  not  a  little  to  prevent  the  com- 
mon study  of  this  elegant  and  useful  branch  of  mathematics. 
This  method  is  not  only  exceedingly  cumbrous,  but  it  has  the 
misfortune  that  its  element,  a  differential  coefficient,  is  a  ratio. 
The  abstract  nature  of  a  ratio,  and  the  fact  that  it  is  a  com- 
pound concept,  pecuharly  unfit  it  for  elementary  purposes.  The 
beginner  will  never  use  it  with  satisfaction,  for  it  does  not  give 
him  simple,  direct  and  clearly  defined  conceptions.  But  while  I 
have  adopted  the  infinitesimal  theory,  I  have  felt  free  to  intro- 
duce the  doctrine  of  Hmits,  and  to  illustrate  and  apply  it.  The 
metaphysical  objections  to  this  method,  if  not  rebutted  by  equal 
difficulties  of  a  similar  character  encountered  in  the  method  of 
Hmits,  are  immensely  overborne  by  its  practical  advantages ; 
for,  let  it  be  remembered  that  no  writer  adheres  to  the  Newton- 
ian method  throughout,  but  ghdes  into  the  other  in  the  Integral 
Calculus,  and  adopts  it  exclusively  in  most  geometrical  and 
physical  applications. 

The  sources  from  which  the  material  has  been  drawn  will  be 
readily  perceived  by  the  mathematician,  and  need  not  be  enum- 
erated here.  That  the  treatise  is  sufficiently  different  from 
others  of  a  similar  purpose  to  justify  its  existence,  the  author 
feels  more  sure  than  that  these  differences  will  commend  them- 
selves to  his  fellow  laborers  in  the  work  of  mathematical  train- 
ing. One  thing,  however,  is  certain,  nothing  in  matter,  arrange- 
ment, or  manner  of  treatment,  has  been  introduced  without 
careful  reference  to  the  capabilities  and  wants  of  such  students 
as  I  have  been  accustomed  to  meet  in  the  class  room  for  more 
than  twenty  years ;  and  few  things  will  be  found  in  the  volume 
but  what  have  been  put  to  the  test  of  class  room  use  many 
times  over. 

A  second  volume,  treating  of  Loci  in  Space,  and  affording  a 
more  extended  course  in  the  Calculus,  will  be  published  as  soon 
as  it  can  be  prepared.  The  present  is  thought  sufficient  for  all 
students  except  such  as  make  mathematics  a  specialty  ;  and  for 
the  latter  the  other  volume  will  be  designed. 


VI  PREFACE. 

In  conclusion  I  must  do  myseK  the  pleasure  to  acknowledge 
mj  indebtedness  to  mj  accomplislied  colleague  and  friend,  Prof. 
J.  C.  Watson,  Ph.  D.,  for  the  original,  direct,  and  simple  method 
of  demonstrating  the  rule  for  differentiating  a  logarithm,  which 
is  given  on  page  25,  and  which  banishes  from  the  Calculus  the 
last  necessity  for  resort  to  series  to  establish  any  of  its  funda- 
mental operations.  I  am  also  indebted  to  my  friend  and  pupil, 
J.  B.  Webb,  B.  S.,  for  many  valuable  suggestions,  and  much  care- 
ful labor  in  reading  both  the  manuscript  and  proof.  To  his 
quick  and  accurate  eye,  and  his  good  taste  and  logical  acumen, 
I  am  indebted  for  the  ehmination  of  not  a  few  defects  which 
might  otherwise  have  disfigured  the  work.  That  there  is  not 
much  of  the  same  sort  of  pruning  yet  needed,  I  have  not  the 
vanity  to  think.  But,  such  as  it  is,  I  commend  my  work  to  the 
consideration  of  teacher  and  student,  with  the  hope  that  it  may 
contribute  to  aid  the  one  in  imparting,  and  the  other  in  acquiring, 
a  knowledge  of  the  elements  of  two  branches  of  science  which, 
in  their  fuller  developments,  exhibit  the  profoundest  and  most 
sagacious  workings  of  the  human  mind,  and  reach  to  the  farthest 
verge  of  the  hitherto  explored  realms  of  human  thought. 

EDWAED   OLNET. 

Ann  Abbob,  Mich.,  July,  1871. 


N.  B. — A  shorter  course  in  tlie  General  Geometry,  ivithout  the 
Calculus,  may  he  taken  from  this  volume  hy  such  as  desire  it. 
For  this  purjoose,  the  first  three  chapters  are  to  he  read,  and  then 
the  course  completed  hy  reading  the  XIV.  and  XV.  Sections  of  Chap- 
ter IV.  If  time  and  purpose  permit,  Articles  {194:,  lOo)  might  he 
read  loith  profit  hy  such  students.  This  ivill  he  found  to  comprise  a 
course  on  Plane  Co-ordinate  Geometry  somewhat  more  full  than  is 
found  in  our  common  text-hooks. 


CONTENTS. 


INTMOnUCTION. 

A  BRIEF  SURVEY  OF  THE  OBJECTS  OF  PURE  MATHEMATICS  AND  OP 

THE  SEVERAL  BRANCHES. 

PAGK 

PuBE  Mathematics. — Definition  (i)  ;    Brandies  enumerated  {2f  3) 1 

Quantity.  — Definition  (4) 1 

NuMBEK. — Definition  {5)  ;    Discontinuous  and  continuous  {6,  7f  8) 2,  3 

Definition    of   the  Several  Branches   of  Mathematics. — Arithmetic    {9 , ; 

Algebra  (JO);  Calculus  {11);  Geometry  {12) \  Descriptive  Geometry  (15)  3,  5 
General  Geometry  divided  into  Two  Books  {14:) 5 


— -tt^^^ 

GENERAL  GEOMETRY 

BOOK  I. 

OF    PLANE    LOCI. 


CHAPTEE  I. 
THE  CABTESIAN   METHOD    OF   CO-OBDINATES. 


SECTION  L 

DEFINITIONS  AND  FUNDAMENTAL  NOTIONS. 

Locus.— Definition  {1) 6 

General  Geometry.  —Definition  {2) 6 

Method  of  Co-ordinates.— What  (5)  ;    Two  Systems  (4) ;    Varieties  of  Eec- 

tUinear  {5) ^'  "^ 

Definitions.— Axes  {6,  7) ;    Origin  {8) ;    Co-ordinates  {9,  10,  11)  ;    Illus- 
tration   « • ' »  " 

Notation.— Of  Co-ordinates  {12)  ;    The  Four  Angles  {13)  ;    Signs   of  the 

Co-ordinates  {14) ^'  ^ 

OnvN-T-rr..-,  —Constant   and  Variable  US'  ;    Definition  of  each  {16,  17    \ 

9 
El  istration  :     Caution.     Sch.  1 


TlU  CONTENTS. 

FAGS 

Indetebminate  Analysis. — ^What,  Sch.  2 9 

To  CoNSTEUCT  AN  EQUATION. — ^What  (^18) 10 


SECTION  IL 

CONSTBTJCTING  EQUATIONS,  OE  FINDING  THEER  LOCI. 

Deitnitions. — A  Continuous  Curve  {19)  ;    Branch  {20)  ;    Symmetry  {21)  ; 

Independent  and  Dependent  Variables  {25) 10-12 

To  LOCATE  A  Point  {22) 10 

To   CONSTBUCT   AN   EQUATION  {23) 11 

Discussing    an    Equation.  —  Wliat  ;     Intersection  ;      Limits  ;     Symmetry 
{26) 12,  13 

"R-yAATPT.-B^S ,^.      11-16 


SECTION  IIL 

THE  POINT  IN  A  PLANE. 

Deitnitions. — ^Equations  of  a  Point  {27) 17 

Equations  op  a  Point. — ^What  {28)  ;    In  different  angles,  In  the  axes,  In 

the  origin,  ScKs.  1,  2  ;  Points,  how  designa'  ed,  Sch.  3 17 

Distance  between  two  Points.  —  General  Formulae   {29) ;    Special  cases, 

Cor.  and  Sch , 18 

Examples.  . , , , , 17,  18 


SECTION  IV. 

THE  EIGHT  LINE  IN  A  PLANE. 

Definition. — Equation  of  a  Locus  {30) 19 

Equations  of  a  Right  Line. —  Through  Two  Points   {31)  ;     Through  One 
Prnnt  {32  i ;   Common  Form  {33) ;   Referred  to  Oblique  Axes  {34:) ;   Meaning 

of  ^,~^„  Cor.l 19-21 

X  — X 

Discussion  oiy  =  ax  -\-  h,  Sch's.  1  and  2 20 

^Methods  of  Constbuctestg  y  =  ax  -\-  h 21 

Locus  of  an  Equation  of  the  Ftest  Degeee  (55) 22 

EXAfcLPLES. 21-23 


SECTION  K 

OF  PLANE  ANGLES,  AND  THE  INTEESECTION  OF  LINES. 

Tangent  op  a  Plane  Angle. — Formulae  for  {36) 23 

Equation  of  a  Line  making  any  given  Angle  with  anothee  Line.  — Com- 


CONTENTS.  IX 

PAGE 

mon  form,  When  passing  through  a  given  point  (57)  ;  When  Parallel  to 
a  given  Line,  Comnion  form,  Passing  through  a  given  point  {38)  ; 
When  Perpendicular  to  the  given  line,  Common  form.  Passing  through  a 

given  poiat  {39) -. 24 

Examples    25,  26 

To  Find  the  Intersection  of  Lines  {40) 26 

ExAMPiiES , 26-28 

Distance  feom  a  Point  to  a  Line  {41)  ;    Between  Parallels,  Cor 28 

Examples 28,  29 


SECTION  VL 

OF  THE  CONIC  SECTIONS. 
Boscovich's  Definition  {4=2) 29 

To   CONSTBTJCT  A   CoNIC   SECTION  {43) 29 

Definitions. — Directrix,  Focus,  Focal  Tangents,  Transverse  Axis,  Conjugate 

Axis,  Latus  Kectum,  Vertices,  Focal  Distances,  Eccentricity  {44) 29,  30 

Axis  of  Hyperbola,  Transverse,  Conjugate,   Conjugate  Hyperbola,  Equi- 
lateral Hyperbola  {47) • 32,  33 

Examples 30-34 

Boscovich's  Ratio  =  Eccentbicitt  {48) 34 

Fundamental  Relations  {45^  46 ,  49) 30,  31,  34,  35 

To  pass  a  Conic  Section  theough  Thkee  Points  {50) 36 

Examples 37 

Equations  of  Conic  Sections. — General  Equation  (5 J? )  ;  Referred  to  their 
Axes,  In  terms  of  A  and  e  {52),  Common  Forms  {53y  54,  50,  57 f  59)  ; 
Referred  to  Axis  and  Tangent   at  Vertex  {55,  56,  57)  \    Of  Conjugate 

Hyperbola  {58) 37-40 

Comparison  op  Equations  of  Ellipse  and  Hypebbola  {60) 41 

Locus  OF  Equation  of  Second  Degree  {61) 41 

Features  op  the  Equation  which  chaeacteeize  the  dippebent  Conic  Sec- 
tions {62)  ;    Species  dependent  on  A,  B,  C,  {63)  ;    All  varieties  included 

in  Aij^  +  Or^  -{-I>y-\-  Ex-j-F=  0  {64) 42,  43 

Examples 42,  43 

Varieties.— Of  Ellipse  {65),  Hyperbola  {66),  Parabola  {67)  ;    Eccentricity 

of  Circle  {68) 43-45 

Examples 46-49 

Exercises  in  producing  various  forms  of  the  equation  of  the  Conic  Sections 

directly  from  the  definition 49-51 

The  Origin  op  the  name  Conic  Section  {69) 51 

Five  Points  in  the  Curve  determine  a  Conic  Section  {70) 52 

Examples 52-54 

Exercises  in  producing  the  equations  of  Conic  Sections  from  various  defini- 
tions.   54-57 


X  CONTENTS. 

SECTION  VIL 

EQUATIONS  OF  HIGHER  PLANE  CURVES. 

PAGE 

Definitions. — Function  {71)',  Classes  ofcT'^);  Algebraic  (75)  ;  Trigo- 
nometrical (7^)  ;  Circular  (.75)  ;  Logarithmic  (7^)  ;  Exponential 
{77) 57 

Loci  Ciassified.— Higher  and  Lower,  Algebraic  and  Transcendental  {78, 70)     58 

Cisson). — Definition  {80)  ;  Construction  {81)  ;  Origin  of  name,  Sch.  1  ; 
Mechanical  method  of  Constructing,  ISch.  2  ;  Equation  of  {82 j  ;  Discus- 
sion of  Equation,  Sch.  1  ;    Duplication  of  cube  by  means  of,  Sch.  2   . . . .  58-60 

Conchoid.  —  Definition  {83)  ;  Construction  {84)  ;  Mechanical  Construc- 
tion, Sch.  ;  Equation  of  {8S)  ;  Discussion  of  Equation,  Sch.  1  ;  Be- 
comes the  equation  of  circle,  Sch.  2  ;  Trisection  of  an  angle  by  means  of, 
Sch.  3 60-62 

Witch.—  Definition  {86)  ;  Construction  {87)  ;  Equation  of  {88)  ;  Dis- 
cussion of  Equation,  Sch 62 

Lemniscate, —Definition  [89)  ;  Construction  {90)  ',  Equation  of  {91)  ; 
Discussion  of  Equation,  Sdi.  1  ;  How  related  to  Equi-lateral  Hyperbola, 
Sch.   2 63 

Cycloid. — Definitions,  of  the  Locus,  Generatrix,  Base,  Axis,  {92,93)  ;  To 
put  the  Generatrix  in  position  {94:)  :  Equations  of  the  Cycloid,  1st  form 
{95),  2nd  form  (96)  ;  Discussion  of  Equation,  Sch.  to  {95),  and  Cor. 
and  Sch.  1  to  (96) 64,  65 

Equations  of  some  i.oci  written  dikectly  feom  the  definitions  {98) ....     66 

NuMBEB  OF  PLANE  CUBVES  INFINITE. — A  fcw  Suggested  {99) 66 


■^♦» 


CHAPTEE  II. 
THE  METHOD    OF  POLAR    CO-ORnU^ATES. 


SECTION  L 

OF  THE  POINT  IN  A  PLANE. 

How  A  Point  is  designated  by  Polar  Co-ordinates  {100) 67 

Definitions. — Pole,  Prime  Radius,  Eadius  Vector,  Variable  Angle,  Polar  Co- 
ordinates ilOl) 67 

Equations  of  a  Point  {102)  ;    Examples 67,  68 

Distance  between  two  Points  {103)  ;    Examples 68 


SECTION  IL 

OF  THE  RIGHT  LINE. 

Equations  op  the  Right  Line. — 1st  form,  2nd  form  {104) ;    Discussion  of 
1st  form,  Sch,  1 ;  Diseussion  of  2nd  form,  Sch.  2  ;  Examples 68-70 


CONTENTS.  XI 

SUCTION  III. 

OF  THE  CIRCLE. 

PAGE 

Equation  "when  the  Pole  is  the  Circumference,  and  the  Polar  Axis  is  a  diame- 
ter {105)  ;    Discussion,  Sch 70,  71 

General  Polab  Equation  {100)  ;  Discussion,  Sch.  ;  Geometrical  Illus- 
tration, ^.5 71-73 

Examples 72,  73 


SECTION  IV, 

OF  THE  CONIC  SECTIONS. 

Polar  Equation  op  Conic  Section  {107)  ;  Of  Parabola  {108)  ;  Of  El- 
lipse and  Hyperbola  {109)  ;  Discussion  of  Equation  of  Parabola,  Sch.  1, 
Of  Ellipse,  Sch.  2,  Of  Hyperbola,  Sch.  3  73-75 

Examples 75,  76 


SECTION  V. 

OF  HIGHER  PLANE  CURVES. 

Polar  Equation  oe  Cissoro  {110)  ;    Discussion,  Sch 76 

PoLAH  Equation  of  Conchoid  {111) ;    Discussion,  Sch 77 

PoLAB  Equation  of  Lemniscate  {112)  ;    Discussion,  Sch 77 

OF   PLANE    SPIRALS. 

Definitions. — Of  Spiral,  Measuring  Circle,  Spire  {113) 77 

Spiral  of  Aechimedes. — Definition  {lid)  ;      Construction  {115)  ;    Equa- 
tion of  {116) 78 

Eecipsocal  or  Hyperbolic  Spibal  (J[j?7)  ;    Equation;    Construction 78 

The  Lituus  {118)  ;    Equation  ;     Construction 79 

LoGAKiTHMic  Spieal  {119)  ',  Definition,  Equation,  Construction  {119) 79 


-♦-♦"^ 


CHAPTEE  III. 
TMANSFOMMATION   OF   CO-ORiyiKATES. 


SECTION  I 

PASSING  FROM  ONE  SET  OF  RECTILINEAR  AXES  TO  ANOTHER. 

Definitions. — Transformation  ;  Two  aspects  of  the  Problem  ;  Primitive 
Axes  or  System  ;  New  Axes  or  System  ;  Illustba.tions  {120)  ;  Practi- 
cal Advan'agec,  Sch 80,  81 


Xii  CONTENTS. 

PAGE 

FoBMULaj  FOB  Passing  from  onb  Eectelixear  Set  of  Axes  to  Anotheb.— 
Ganeral  Formulae  {122)',  From  aay  set  to  a  Parallel  set  {123);  From 
Eectangular  to  Oblique  {124);  From  Rectangular  to  Rectangular  {125); 
From  Oblique  to  Rectangular  {126);  The  foregoing  where  the  origin  is 
unchanged  {127);  From  Oblique  to  Rectangular,  when  a  and  a'  sig- 
nify the  angles  which  the  Oblique  or  Primitive  axes  make  with  the  Rect- 
a  igular,  or  New  axis  of  x,  Sch.. . 82-84 

EXAMPIIES 84-90 


SECTION  IL 

PASSING  FEOM  RECTILINEAR    TO  POLAR  CO-ORDINATES, AND  VICE  VERSA., 

Formula  fob  passing  from  Rectilineab  to  Polar  {128) 90 

Fobmuils;  fob  passing  from  Polab  to  Rectilineae  {129) 91 

ExAMPiiES « 91,  92 

^-♦-> — 


CHAPTER   lY. 

TBOPERTIES   OF  PLANE    LOCI  INVESTIGATED   BY 
3IEANS   OF   THE  EQUATIONS  OF   THOSE  LOCI. 


SECTION  I 

TANGENTS   TO    PLANE  LOCI. 

(rt)    BY    RECTILmEAR    CO-ORDINATES. 

Definitions. — Consecutive  points  {130);  Tangent  {131);  Tangent  has 
the  same  direction  as  the  Curve,  Cor.  {132) 93 

(111 
Geometricai.  Signification  of  -^  {133);    A  Tangent  which  makes  any 

dx 

given  angle  with  the  axis  of  x,  \Vhica  is  parallel,  Which  is  perpendicular 
yl34:);    Signification  of  ~  when  the  axes  are  oblique  {135);  Examples.  93-96 

Equations  of  Tangenis.  — Gensral  Equation  {136)  ;  Of  the  Ellipse,  The 
Hyperbola,  The  ParaboU,  The  Circle?,  and  other  Examples  ;  The  Intercepts 
of  the  Axes  by  a  tangent  {137),  Wi  h  the  axis  of  x  in  Ellipse,  Hyperbola, 
Parabola  ;  Other  Examples  ;  To  draw  a  tangent  to  an  Ellipse  {138),  To 
an  Hyperbola  {139),  To  a  Parabola  {140) 96-101 

SuBTANGExVTS.  —Definition  {141}  ;  General  value  of  {142}  ;  Of  an  Ellipse, 
Hyperbola,  Parabola,  other  Examples  ;     Use  in  drawing  tangents 101,  102 

Length  of  Tangent.— General  formula  {143) ;  Of  an  Ellipse,  Hyperbola, 
Parabola •  102,  103 

Asymptotes  (rectilinear).— Definition  {144\  Illustrations;  To  examine  a 
curve  for  A.sr/mptoies,— General  Method  (145  ,  By  Inspection  {148),  By 
Develop^no:  the  function  149  ;  An  Asymptote  the  limiting  position  of  a 
Tangent  {146   ;     Equation  of  (i^T*   ;    Examples  103-107 


CONTENTS.  XIU 

PAGS 

(6)  TANGENTS  TO  POLAB  CURVES. 

How  Deteemi»ed  {150) 107 

SuBTANGENT.— Definition  {151)  ;    General  Value  {152) ;    Examples. ..  107-109 
Asymptotes. — How  Determined  {153) ;    Examples 109,  110 


SECTION  11. 

NOEMALS  TO  PLANE  LOCI. 

(a)    BY   RECTANGULAR   CO-ORDINATES. 

Definition  or  Nobmal  {154:) HO 

General  Equation  (155).— Signification  of  —  —  {156)  ;    Normal  to  El- 
lipse, Hyperbola,  Parabola,  and  otber  Examples  110,  111 

Subnormal.— Definition  {157)  ;    General  Value  {158)  ;    To  Cycloid  {159)  ; 
To  draw  a  Tangent  to  the  Cycloid  {160)  ;    To  draw  a  Tangent  making  a 

given  angle  {161 )  ;    Examples HI,  112 

Length  of  Noemal  {162)  ;    Examples 112 

Pekpendiculaii  upon  a  Tangent   {163)  ;    From  the  focus  of  a  Parabola 
{164)  ;    Examples • 112,  113 

(&)    NORMALS   TO    POLAR   CURVES. 

SuBNOEMAii. — ^Definition  {165}  ;     General  Value  {166)  ;    Examples  ...  113,  114 

Length  of  Normal  to  Polar  Curve  {167) H^ 

Length  of  Perpendicular  from  the  Pole  upon  the  Tangent  op  a  Polar 
Curve  {168) H^ 


SECTION  IIL 

DIRECTION  OF  CURVATURE. 

(a)    BY   RECTANGULAR    CO-ORDINATES 
Criteria  for  determining  Direction  of  Curvature. 

Sign  of    p{    {169)  ;    Sign  of   ^'   {170)  ;     Sign   of  yp{  {171)  ;    Ex- 
dxr  ay-  ax^ 

amples 114r-116 

{h)    BY   POLAR   CO-ORDINATES. 

Definition  op  Direction  of  Curvature  of  Polar  Curves  {172) 116 

Ceitebia  for  determining  {17 3 f  174)  ;    Examples 116,  117 


SECTION  IV. 


SINGULAR  POINTS. 


Definition  and  Enumeration  {175) 117 

Maxima  and  Minima   Ordinates.— Definition  {176);    To  determine  their 


XIV  ^  CONTENTS. 

PAOS 

position  and  value  {177) ',    A.  negative  maximum  or  minimum  {178) ; 

Examples 118,  119 

Points  of  iNFiiExioN. — Definition  {170),  Illustration  ;  How  determined  by 
Kectangular  Co-ordinates  {180)  ;  By  Polar  Co-ordinates  {181)  ;  Ex- 
amples     119-121 

Multiple   Points. — Definition,   Species  {182)  ;     How  determined  {183)  ; 

Examples 121-123 

Cusps. — Definition,  Kinds  {184)  ;    How  determined  {185)  ;    Examples  124,  125 
Conjugate  Points. — Definition  {186)  ;     Two  Criteria  {187 >  188)  ;    How 

to  examine  a  curve  for  Conjugate  Points  {180)  ;    Examples. , 125-127 

Shooting  Points. — Definition  {100)  ;    Examples 127,  128 

Stop  Points. — Definition  {101) ;    Examples 128 


SECTION   K 

TRACING  CURVES. 

Definition  (102) 129 

General  Method  {103,  and  Sch.)  ;    Examples. . , 129-132 

To  tbace  a  cubve  of  the  Second  Okdee.  — By  direct  inspection  of  its  equa- 
tion   {104:)  ;    Examples ;    By    Transformation  of    Co-ordinates  {105)  ; 

Examples 132-135 

To  Tkace  a  Polab  Cubve  {106) ;    Examples 135-137 


SECTION  VL 

RATE   OF  CURVATURE. 

Definitions. — Curvature  {107),  Illustration  ;  Osculatory  Circle  {108),  Il- 
lustration ;  Radius  of  Curvature,  Centre  of  Curvature  {100)  ;  Parameter 
{202) 137-140 

Contact. — What,  How  closeness  of  Contact  is  characterized,  Orders  of  Con- 
tact {200),  Geometrical  Illustration  {201)  ;  Order  of  Contact  dependent 
upon  Parameters  {203)  ;  Order  of  Contact  of  Eight  Line  {201',  Of 
Circle,  Of  Parabola,  ElHpse,  Hyperbola  {205,  206)  ;  Eestriction  of 
these  statements  {207)  ',  Contact  of  a  Eight  Line  at  Point  of  Inflexion 
{215)  ;  Contact  of  Osculatory  Circle  at  points  of  Maximum  and  Mini- 
mum Curvature  {216),  At  the  Vertices  of  the  Conic  Sections  {217)  ■  ■  139-146 

Eadius  of  Cuevatuee.  —General  Formula  in  terms  of  Eectangular  Co-ordi- 
nates {208)  ;  Signification  of  the  sign  {200)  ;  Eadius  of  Curvature  of 
the  Conic  Sections,  At  the  vertices  {210,  212),  Varies  how  {211,  213)  ; 
Centre  of  Curvature  in  the  Normal  {214)  ;    Examples 141-145 

When  Osculatory  Curves  intersect  and  when  not  (218)  ;  When  the  Os- 
culatory Circle  Cuts  a  Conic  Section  (210) 146,  147 

Eadius  op  Curvature  of  Polar  Curves  {220)  ;  Involving  the  Normal 
(^221)  ;    Examples 147,  148 


CONTENTS,  XV 

PAGX 


SECTION  VIL 

EVOLUTES  AND  INVOLUTES. 

Definition  (222),  Illustration 148 

To  FIND  THE  EVOLUTE  {223)  ',    Examples  ;    The  Evolute  of  a  Cycloid  an 

Equal  Cycloid  {225)  ;    Same  Geometrically,  ^eh 149-151 

NoBMAii  TO  Involute  Tangent  to  Evolute  {220) 151 

Kadius  of  Cubvatuee  vabies  as  Aug  of  Evolute  {227) 151 

A  CURVE  desckibed  mechanically  feom  its  Evolute  {228) 152 

a  curve  has  but  one  evolute,  but  an  evolute  has  an  infinite  numbeb  of 

involutes  {229) 152 


SECTION  VIIL 

ENVELOPES  TO  PLANE  CURVES. 

Definition  {230),  Illustration 152,  153 

To  find  the  Envelope  {231)  ;    Examples 153-159 

Envelope  tangent  to  the  inteesecting  seeies  {232) 154 

Caustics. — General  Equation  {233)  ;  When  the  incident  rays  are  parallel 
to  the  axis  of  the  reflector  {23d),  When  perpendicular  {235) ;  Illustra- 
tion ;    Examples 156-159 


SECTION  IX, 

RECTIFICATION  OF  PLANE  CURVES. 

Definition  {237) 159 

By  Eectangulae  Co-oedinates. — General  Formula  {238)  ',  Examples ; 
Circumferences  of  Circles  are  to  each  other  as  the  radii  {240)  ;  Value  of 
7t  [24:1 )  ;  Arc  of  Cycloid  equals  twice  the  corresponding  chord  of  the  gen- 
eratrix i24:2) 159-162 

By  Polab  Co-obdinates. — General  Formula  [24:3)  ;    Examples 163,  164 


SECTION  X. 

QUADRATURE  OF  PLANE  SURFACES. 

Definition  {244:) 164 

By  Kectangulab  Co-oedinates.— General  Formula  {245) ;  Examples  ;  Areas 
of  Circles  to  each  other  as  squares  of  radii  [246)  ;  Area  of  Circle  whose 
radius  is  1  {247)  ;  Area  of  Circle  =  ^r  X  circumference  {248)  ;  Area  of 
Segment  of  Circle  (249)  ;  Area  of  Ellipse  compared  with  Circumscribed 
and  Inscribed  Circles  (250) 164-168 

B^  PoLAE  Co-ordinates.— General  Formula  ^251)  ;    Examples 168,  169 


XVl  CONTENTS. 


SECTION   XI 

QUADRATURE  OF  SURFACES  OF  REVOLUTION. 

Definition  {252)  ;    Illustrations l(il) 

I^ENEEAL  FoKMULA  {233)  ',  Examples  ;  Surface  of  a  Sphere  =:  4  great  Cir- 
•  cl.-s,  or  Circumference  X  Diameter,  Cor.  1 ;  Area  of  Zone  {254:) ;  Sphere 
/    and  Circumscribed  Cylinder,  Sch 169-170 


SECTION  XII 

CUBATURE  OF  VOLUMES  OF  REVOLUTION. 

Geneeal  Foemula  {255)  ;  Examples  ;  Volume  of  a  Sphere  =  the  surface 
X  3  radius  {256)  ;  Volumes  of  Spheres  are  to  each  other  as  the  cubes  of 
their  radii  {257)  ',  Volume  of  a  Segment  {258)  ;  Volume  of  Sphere  and 
Circumscribed  Cylinder  {259) 171,  iTa 


SECTION  XIII 

EQUATIONS  OF  CURVES  DEDUCED  BY  THE  AID  OF  THE  CALCULUS. 

Teacteix.  — Dehuition  {200)  ;    Equation  (261) 172,  173 

Locus  WHOSE  SUBNOEMAL  IS  CONSTANT  [262) 173 

Locus  WHOSE  NOEMAL  IS  CONSTANT  {263) 174 

Locus  WHOSE  SUBTANGENT  IS  CONSTANT  {264) 174 

Locus  WHOSE  SuBNOEMAL  VAEIES  AS  THE  SqUAEE  OF  ITS  AbSCISSA  {265) 174 

Locus  WHOSE  Area  is  twice  the  peoduct  of  its  Co-oedinates  {266) 174 

Locus    WHOSE    AeC   VAEIES   AS   THE    SQUAEE    EOOT    OF    THE    THIED    POWEE  Ol'  I'rS 

abscissa  {267) 174 


SECTION  XIV. 

OF  TANGENTS  AND  NORMALS. 

[WITHOUT   THE   AID    OF   THE   CALCULUS.] 

Tangents. — General  Method  of  producing  the  equation  of  {268)  ;  Ex- 
amples,— Tangent  t  >  Parabola,  Ex.  1  ;  Elhpse,  Ex.  4  ;  Hyperbola,  Ex.  10  ; 
Tangent  of  the  angle  which  a  tangent  to  a  Conic  Section  makes  with  the 
axis  of  X  {260),  Examples  ;  To  find  the  point  on  a  curve  from  which  a 
tangent  must  be  drawn  to  make  a  given  angle  with  the  axis  of  x,  be  paral- 
lel, be  perpendicular  {270)  ;    Examples 175-179 

SuBTANGENTs.— Definition  {271)  ;  To  find  the  length  of  {272)  ;  Ex- 
amples,— Subtangent  in  Parabola,  To  draw  a  tangent  by  means  of  {273)  ; 
Subtangent  of  Ellipse,  To  draw  a  tangent  by  means  of  {274,  275) ;  Sub- 


CONTENTS.  XVll 

PAGE 

tangent  of  Hyperbola  {270),  To  draw  a  tangent  by  means  of  {277)  ',  Half 
either  axis  a  mean  proportional  between  its  intercepts  by  a  tangent  and 
ordinate,  Ex.  4  ;  Analogy  between  the  equations  of  the  Conic  Sections  and 
the  equations  of  their  tangents  {278) 179-181 

Normals. — Definition  {279)  ;    To  produce  the  Equation  of  Normal  {280)  ; 

Tangent  of  angle  which  Normal  makes  with  axis  of  x  {281)  ;    Examples, 

Normal  to  Ellipse,  Hyperbola,  Parabola,  Circle  ;    Expressions  for  tangent 

*     of  the  angle  which  a  Normal  to  a  Conic  Section  makes  with  the  axis  of  x 

{282) 181,  182 

SuBNOEMAiiS. — Definition  {283)  ',  Examples  in  the  Conic  Sections,  Is  con- 
stant in  the  Parabola  and  =  p,  To  draw  a  tangent  by  means  of  the  latter 
property.  Ex's  1  and  2 183 

The  Peepediculae  feom  the  focus  or  a  Paeabola  upon  the  tangent  {284:)  ; 
Cor.  {285)  ;  To  find  the  focus  of  a  Parabola  when  the  curve  and  its  axis 
are  given,  Also  to  draw  a  tangent  {280 f  287 ?  288) 183 


SECTION  XV, 

SPECIAL  PROPERTIES  OF  THE  CONIC  SECTIONS. 

Eadh  Vectoees. — Definition  {289)  ;  Sum  of  in  Ellipse  and  diflference  in 
Hyperbola  {290)  ;  Length  of  each  {291)  ;  To  construct  an  EUipse  and 
Hyperbola  on  this  principle  {292) ;  Eadii  Vectores  make  equal  angles  with 
the  tangent  in  Ellipse  and  Hyperbola  {293)  ;  Corresponding  property  in 
Parabola  {290)  ;  Angles  included  by  the  Eadii  Vectores  and  Normal,  in 
Ellipse  and  Hyperbola  {294:) ;  To  draw  a  tangent  upon  these  principles,  1st, 
from  a  point  in  the  curve,  2nd,  from  a  point  without  {295) ;  Same  problems 
in  reference  to  the  Parabola  {298) 184-187 

The  eectangle  of  PEEPENDicuiiAES  FEOM  FOCI  UPON  Tangent  {299) 187 

The  Semi-conjugate  axis  a  mean  peopoetional  between  focal  distances 
{300) 187 

Supplementaey  Choeds  and  Conjugate  Diameters. — Definition  of  Ordinate 
{301),  Of  Supplementary  Chords  {302),  Of  Conjugate  Diameters  {303)  ; 
Fundamental  property  of  Supplementary  Chords  {304,  305)  ;  When 
drawn  on  the  Conjugate  Axis  {300)  ;  When  drawn  from  a  point  in  the 
Conjugate  Hyperbola  {307)  ',    This  property  in  the  Circle  {308)  ;    Paral- 

lelism  of  Sup.   Chords  to  the  axes  {309)  ;    The  —  sign  m  axi'  = j 

{310)  ;  Discussion  of  the  Angle  included  by  Sup,  Chords  {311) ;  Sup. 
Chords  parallel  to  Tangent  and  Diameter  {312 ,  313)*;  To  draw  Tangents 
by  this  property  {314)  ;    Eelations  between  Conjugate  Diameters  and  the 

Axes  {318)  ;    Examples 188-194 

Oedinates. — Eelation  to  each  other,  in  Ellipse  {319),  in  Hyberbola  {322), 
in  Parabola  {333)  ',  Corresponding  properties  of  oblique  ordinates  {325) ; 
Eelation  of  an  ordinate  to  the  corresponding  segments  of  its  diameter 
{320)  ;  Latus  Eectum  a  third  proportional  to  the  axes  {321 )  ;  The  rela- 
tion of  ordinates  in  the  circle  {323)  ;  Eelation  of  ordinates  to  the  conju- 
gate axis  of  Ellipse  {324)  ;  Parallel  chords  bisected  hj  Diameter  {320 , 
334)  ;    To  find  the  centre,  axes  and  foci  of  a  Conic  Section  when  the  curv- 


XVIU  CONTENTS. 

PAGB 

ature  is  given  {327 f  335)  ;  Ordiuates  of  different  Ellipses  on  same  axis 
{Ji28);  Of  Ellipse  and  Circle  on  same  axis  {329)  ;  The  Trammel  {330); 
Ordinates  to  different  ellipses  on  same  Conjugate  Axis  {331)  ;  Of  Ellipse 
and  Inscribed  Circle  {332) 195-199 

EcjsNTEic  Angle. —Definition  (336)  '■  Sine  and  cosine  of  this  angle  {337); 
Advantages,  Sch. ;  Equation  of  Tangent  to  Ellipse  in  terms  of  this  angle 
{338)  ;  Eccentric  angles  of  the  vertices  of  the  Conjugate  Diameter 
{339}  ;  To  draw  a  Conjugate  Diameter  on  this  principle  {340)  ;  Kect- 
angle  of  Kadii  Vectors  =  Square  of  Conjugate  Diameter  {341)  ;  Sum  of 
the  Squares  of  Conjugate  Diameters  constant  {342)  ;    Examples 199-201 

The  Intercepts  or  a  Secant  between  the  Htpeebola  and  its  Asymptotes 
{343) ;    To  construct  an  Hyperbola  on  this  principle,  8ch 201,  SOS 

Parajmetee  to  ai^  Diameter  of  a  Conic  Sectiok. — Definition  (344) ; 
Distance  from  point  in  Parabola  to  focus  {345) ;  Parameter  to  any  dia- 
meter of  Parabola  {340);  Parameter  to  any  diameter  of  Parabola  a 
double  ordinate  through  focus  {347) ;  Chord  of  Ellipse  through  focus 
{348) ;    Sch.  {347)  not  applicable  to  Ellipse  {349) 202,  203 

Chord  of  Curvature. — Definition  {350) ;  In  the  Parabola  chord  of  cur- 
vature through  focus  a  parameter  to  the  diameter  through  potut  of  con- 
tact {351} ;  Intercept  on  this  diameter  by  the  osculatory  circle  equals 
this  chord  of  curvature  {351) 203,  204 


CONTENTS.  XIX 


THE 


INFINITESIMAL    CALCULUS. 


*♦* 

INTnonUCTION. 

PAGE 

Definitions.  —Quantity  {!) ;  Number  {2) ;  Discontinuous  and  Continuous 
Number  {3,  4:,  5),  Illustrations  ;  An  Infinite  Quantity  (6) ;  An  Infini- 
tesimal (7) ;     Caution  {8). 1,  2 

Infinites  and  Infinitesimals  eecipeocals  of  each  other  (9,  10) 3 

Obders  of  Infinites  and  iNFiNiTESiMAiiS.— What  {11}  ;  Belations  to  eacb 
other  {12) 3 

Axioms  {IS,  14:,  15,  10,  17,  18)  ;     Illustrations  ;    Examples 4,  6 

Constants  and  Variables. — What  {19,20);  Any  expression  containing  a  va- 
riable is  a  variable  when  taken  as  a  whole  {21)  ;  Distinction  of  Depend- 
ent and  Independent  Variables  {22,  23,  24),  Illustration  ;  Equicrescenfc 
Variable  {2S)  ;     Contemporaneous  Increments  {20}  ;    Illustration 6,  7 

Functions  and  their  Forms. — Definition  of  Function  {27),  Illustration  ; 
Exact  limitation  of  the  term,  Sch.  ;  Functions  classified  as  Algebraic  and 
Transcendental,  and  the  latter  as  Trigonometrical,  Circular,  Logarithmic 
and  Exponential,  with  Definitions  {28,  29,  30,  31,  32,  33)  ;  Functions 
Explicit  or  Implicit  {34,  35,  36),  Notation  {37)  ;  Functions  Increasing 
or  Decreasing  {38,  39,  40)  7-9 

The  Infinitesimal  Calculus. — What  {41) t  Illustration;  Two  Branches 
{42) , 9,10 

^»» 


CHAPTER  I. 

TSE  DIFFERENTIAL    CALCULUS, 


SECTION  L 

DIFFEEENTIATION  OF  ALGEBRAIC  FUNCTIONS. 

Definitions. — The  Differential  Calculus  {43)  ;    A  Differential  {44)  ;    Con- 
secutive Values  {45),  Illustrations 11 

Notation  for  a  Differential  {4G) 11 

EULES   FOR   DIFFERENTIATING  ALGEBRAIC  FUNCTIONS. 

EuLE  1.— To  Differentiate  a  Single  Variable  {47),  Geometrical  Illustration    12 

Bule  2. — Constant  Factors  {48),  Geometrical  Illustration 12,  13 

BuLE  3. — Constant  Terms  (49) ;    Geometrical  Illustration  ;    An  infinite 

variety  of  functions  may  have  the  same  differential  {50) 13 

BuLE  4. — The  Sum  of  Several  Variables  (51^.  Illustration  ;    Character  of 

dr,  dy,  dz.  etc  ,  Sch 14 


XX  CONTENTS. 

PAGE; 

Rule  5. — The  Product  of  Two  Variables  (52),  Illustration  ;    Rate  of  Change    14 

Rtile  6. — The  Product  of  Several  Vaiiables  (33) 15 

RT7ii£  7. — Of  a  Fraction  with  variable  numerator  and  denominator  (54)  , 
With  constant  numerator  (55)  ;    With  constant  denominator, 

Sch 15,  16 

Rule  8. — Of  a  Variable  with  exponent  {36)  :     Square  Root  (57)  ;     Other 

special  rules,  Sch 16 

ExEECiSES  in  differentiating 16-22 

Tt.t.ustbative  Examples  showing  the  significance  of  differentiation 22-25 


SECTION  11. 

DIFFERENTIATION  OF  LOGARITHMIC  AND  EXPONENTIAL  FUNCTIONS. 

Definition. — Modulus  {S8)  25 

To  DiEEEBENTiATE  A  LoGABiTHM,  Common,  Napierian  {SO) 25 

To  DiFFEBENTiATE  EXPONENTIALS. — With  Constant  base  {00),  With  reference 
to  Napierian  logarithms  {01)  ;  When  the  base  of  the  Exponential  is  the 
base  of  the  system  of  logarithms  {62)  ;     Of  exponential  with  variable  base 

(63) 26 

Exercises 27,  28 

DrFEEKENTIATING  A  VARIABLE  WITH  TmAGINABY  EXPONENT  {6S) 28 

IujUSTBAtive  Examples  28-30 


SECTION  IIL 

DIFFERENTIATION  OF  TRIGONOMETRICAL  AND  CIRCULAR  FUNCTIONS. 

Of  Teigonometeical  Functions. — Of  a  sine  {66)  ;    Of  a  cosine  {67),  Signifi- 
cance of  the  sign  {68) ;    Of  a  tangent  {69) ;    Of  a  cotangent  (70) ;    Of       / 
a  secant  {71)  ',    Of  a  cosecant  {72)  ;    Of  a  versed-sine  {73)  ;    Of  a  co-       ' 

versed-sine  (74) 30-32 

Exercises 32 ,  33 

Illustbative  Examples 34,  35 

Of  Cieculab  Functions. — In  terms  of  sine  (75),  Relation  io  differentiating 
trigonometrical  functions  {76)  ;  In  terms  of  cosine  {77)  ',  In  terms  of 
tangent  {70)  ;  In  terms  of  cotangent  {80)  ;  In  terms  of  secant  {81);  In 
terms  of  cosecant  {82)  ;  In  terms  of  versed-sine  {83)  ;  In  terms  of  co- 
versed-sine  {81:) 35,  36 

Exercises  ;    Geometrical  Illustration 37-39 


SECTION  IV, 

SUCCESSIVE  DIFFERENTIATION  AND  DIFFERENTIAL  COEFFICIENTS. 

Successive  Differentiation. — Definiiions — Of  successive  differentials  {87), 
IllustraMons ;  Of  Second,  Third,  etc.,  differentials  {88}  ;  To  produce 
cnccossivc  diff^reiitial^  (89) 40,  41 


CONTENTS.  Xxi 

PAGR 

EXEECISES     41 ,  42 

DuTEEENTiAii  CoEPFiciENTS. — A  first,  A  second,  A  third  {90)  ;    Illustration  ; 

Differential  coefficients  generally  variable,  Sch 42-44 

Exercises  .......,,,,,,,,,,,,,,, 43,  44 


SECTION   V. 

FUNCTIONS  OF  SEVERAL   VARIABLES,   PARTIAL  DIFFERENTIATION,  AND 
DIFFERENTIATION  OF  IMPLICIT  AND  COMPOUND  FUNCTIONS. 

Functions  oi'  Independent  and  of  Dependent  Vabiables  {91)  ;    lUustra-        j 
tions 44 

Definitions.— Partial  Differential  (92)  ;  Total  Differential  (93)  ;  Illustra- 
tions ;  Partial  Differential  Coefficient  {94)  ;  Total  Differential  Coefficient 
{95)  ;    Equicrescence  of  variables,  Sch 45 

Total  Deffseential  equals  the  Sum  or  the  Paetial  Diffeeentials  {97)  ; 
Illustrations  ;    Exercises 45-48 

Notation  of  Differential  Coefficients  {98) 48 

Total  Diffeeential  Coefficient. — Of  function  of  two  variables,  Formula, 

(99 )  ;    Meaning  of  -i-  in  sucb  cases,  and  distinction  between  I  —    and  — , 
I  ^       dx  [dxj         dx 

i  Sch.  ;     Of  three  variables,  Formula  {100)  ;     "When  u  =  f{y,  z,  w),  and 

J  y  =  q){x),  z  =  cpiix),  and  w  =  qj^ {x)  {101)  ;    Exercises 48-50 

Implicit  Functions. — To  differentiate  f{x,  y)  =  0  {102)  ;     Why    —  p=  0, 

and  —J  or  —  not,  Sch. ;    Exercises 51,  52 

dx        dy 

Compound  Functions. — Definition,  and  methods  of  expression  {103)  ;  To 
differentiate  u=f{y),  when  y  =  cp{x)  {104:);  Exercises;  To  differen- 
tiate u  =  cp{z),  when  z  =f{x,  y)  {103) 52,  53 


k 


SECTION  VL 

SUCCESSIVE  DIFFERENTIATION  OF  FUNCTIONS  OF  TWO  INDEPENDENT 
VARIABLES,  AND   OF  IMPLICIT  FUNCTIONS. 

Both  Vaeiables  may  be  Equiceescent  {106)  ;    Illustration 53 

Successive  Paetial  Diffeeentials. — ^Definition  {107)  '■>  Notation  {108)  ; 
Partial  Differential  Coefficients,  Sch.  ;  Examples  ;  Order  of  differentiation 
■unimportant  (109)  ;  Examples  ;  To  form  successive  Partial  Differentials 
of  a  function  of  two  Independent  Variables  {110)  ',  Law  of  the  formula, 
Sch 54-58 

To  FOEM  Successive  Diffeeential  Coefficients  of  an  Implicit  Function  of 
A  Single  Vaeiable  {111)  ;    Examples 58-60 

Deeived  Equations. — What;  Orders  of;  First  and  Second  produced  from 
u  =  0  =f{x,  y)  {112) 60,  61 


XXU  CONTEMS. 


SECTION  VIL 

CHANGE  OF  INDEPENDENT  VAEIABLE. 

PAGB 

Why  necessaet  {113) 61 

FoBMS   OS  -^,  -~,  -— ,  -vrlien  neither  variable  is  equicrescent  {114:)  ;    Ex- 

(XX     CuC       CfriC 

amples . .    62,  63 

FoBMUIiaJ  FOE  CHANGING  FROM  X  10  p  {116) 64 

FoEMuxLiE  FOE  Inteoducing  A  NEW  VAEiABiiE  6  as  the  equicrescent  {117) 64 

Examples 65,  66 

To  Express  the  Partial  Duteeential  Coefficients  of  u  =f{x,  y),  in  terms 
of  r,  and  6,  when  x  =  cp{r,  6),  and  2/  =  <p.  {r,  Q)  {118) ;  When  x  =  r  cos  6, 
and 2/  =  rsin 0  {119) 66 


■^  »» 


CHAPTER  11. 

ArjPLICATIONS  OF  THE  DIFFERENTIAL  CALCULUS, 


SECTION  L 

DEVELOPMENT  OF  FUNCTIONS. 

Definition. — Development,  what  {120)  \    Illustration 67 

Maclaubin's  Foemula. — Object  of  {121)  ;  The  Formula  produced  {122)  ; 
How  to  repeat  it  {123)  ',     Other  forms  of  writing  it  {124}  ;    Exercises  in 

appUcation 67-73 

To  Deduce  the  Binomial  Foemula,  £k.  2 69 

The  Theoey  of  Logaeithms. — lo  Produce  the  Logarithmic  Series, — General, 
Kv.  7,  Napierian  {120)  ;  Adapted  to  computation  {127)',  Relation  of 
logarithms  of  the  same  number  in  different  systems  {128)  ',  To  find  the 
Modulus  of  the  Common  System  {129)  ;  To  compute  a  table  of  common 
logarithms  {130);  To  find  the  relation  of  the  modulus  to  the  base,  Ex.  8  ; 
To  find  the  Napierian  base  {131)  ;  To  produce  the  exponential  series, 
Ex.  9  ;    To  obtain  the  Napierian  base  from  the  exponential  series  {132) . .  70-72 

To  FIND  the  value  OF  7t  from  y  =  tan—'  x,  Ex,  10,  and  Sch 73 

Maclauein's  Foemula  not  Applicable  to  all  forms  of  functions  {133) ;  Ex- 
amples ;    Occasion  of  the  inapplicability  {134) 73,  74 


Tayloe's  Foemula.— Object  of  {135) ;    Lemma,  If  w  =/(«  +  ?/),  -5-  =  j- 

{136)  ;  Coefficients  depend  upon  the  form  of  the  function,  Sch. ;  Ex- 
amples ;  Formula  Produced  {137)  ',  How  to  state  it  {138)  ;  Exercises 
in  applying  it ;    Use  in  developing  y  =f{x),  when  x  takes  an  increment 

{139) 74-78 

Tayloe's  Foemula  sometimes  fails  for  certain  values  of  the  variable  {140) 
Examples  ;  Distinction  between  the  failure  of  Maclaurin's  and  of  Taylor's 
formulae,  Sch.  2 78,  79 


CONTENTS.  xxiii 


SECTION  IL 

EVALUATION  OF  INDETERMINATE  FORMS. 

PAGK 

The  Indeterminate  Fobms  Enumerated  {14:1)  ;  Illustration  ;  All  reducible 
to  one  ;  From  what  the  apparent  indetermination  sometimes  arises  {14:2) ; 
Illustration 80 

Evaluation  oe  ^  {143) ;    Examples 81,  82 

Evaluation  oe  —  {144)  ;     Examples 83,  84 

00 

Evaluation  of  0  X  oo  {145)  ;    Examples 84,  85 

Evaluation  of  oo  —  oo  {146) ;    Examples , , 85,  86 

Evaluation  op  Qo,  ooo,  and  1"  {147)  ;    Examples 86,  87 


SECTION  III, 

MAXIMA  AND  MINIMA  OF  FUNCTIONS  OF  ONE  VARIABLE. 

Definitions. — Maximum  {148)  ;  Illustrations  ;  Minimum  {149)  ;  Illus- 
trations ;  Same  function  may  have  several  maxima,  and  several  minima 
values  {150)  ;  Geometrical  Illustrations ;  Technical  use  of  the  terms 
{151) 88,  89 

dxi 
At  a  Maximum  oe  Minimum  -^  =  0,  or  oo  {152)  ;    Geometrical  Illustrations  ; 

dv 
Not  all  values  arising  from  -^  =r  0  or  oo,  correspond  to  maxima  or  min- 

ima 89-91 

Sign  op  -^  at  points  of  Maxima  and  Minima  (153) 91 

dx^  "  ' ' 

Ordinaey  Method  op  Pboceduke  {154)  ;    Axioms  which  facilitate  the  work 

{155)  ;    Examples  ;     Failure  of  the  ordinary  method,  Ex.  10  ;    Method 

in  such  cases,  Ex.  10,  and  {156 )  ;    Geometrical  Problems 91-101 

-4-^^ 


CHAPTER   III. 

THE  INTEGRAL    CALCULUS. 


SECTION  I 

DEFINITIONS  AND  ELEMENTARY  FORMS. 

Definitions.— The  Integral  Calculus  {157)  ;  An  Integral  {158) ;  Integra- 
tion {159)  ;  Sign  of  Integration  {160)  ;  Illustrations  of  preceding  defi- 
nitions       102 

No  such  thing  as  a  Pkocess  of  Integration  {161) 103 

Theee  Elementaey  Peopositions.— Constant  factors  {162)  ;    Algebraic  sum 


Xxiv  CONTENTS. 

PAGE 

of  differentials  {163)  ;    The  Indeterminate  Constant  {164:) ;    Method  of 

disposing  of  this  constant,  8cli 102,  103 

Two  EiiEMENTABY  KuiiES.— For  a  monomial  differential,  or  a  differential 
which  may  be  so  treated  {165)  ;  Examples  ;  For  differential  of  a  loga- 
rithm {166) 104 

Ei:ementaey  Fobms  {167) 1^^ 

Stjbokdinate  Elementary  Fokms  {168)  ',    How  obtained 106,  107 

LoGAEiTHMic  Teigonometeical  Foems  {169) 108 

Examples  in  application  of  Elementary  Forms 108-117 

Simple  Expedients  for  reducing  to  elementary  forms. — Introduction  of  a 
constant  factor  {170),  and  Ex's  following  ;  Caution,  Sch.  ;  Reduction  of 
a  differential  by  transfer  of  variable  factor  {171)  ',  How  the  constant  is 
written  in  logarithmic  integrals,  Sch 108-117 


SECTION  IL 

RATIONAL  FRACTIONS. 

Detinition. — Eational  Fraction  {172) 117 

NUMEEATOB  MAY  BE  MADE  OF  LOWEE  DIMENSIONS  THAN  THE  DeNOMINATOB  {173)   118 

Separation  into  Parts  by  Indeterminate  CoefScients. — When  the  Denomina- 
tor is  resolvable  into  Beol  and  Unequal  factors  {174)  ;  When  resolvable 
into  Real  and  Equal  factors  {175)  ;  When  the  factors  are  Imaginary,  Sch. ; 
When  the  factors  are  Eeal,  Equal  Quadratics  {176)  ;  How  to  find  the  fac- 
tors {177)  ;  Forms  on  which  the  integration  depends  in  these  cases 
{178) ;    Examples 118-123 


SECTION  IIL 

RATIONANIZATION. 

Binomial  Dutebentials.— Reducible  to  the  form  x'^{a  -(-  ho(^)Pdx  {180) ; 
Can  be  rationalized,  1st,  {181)  ;  2nd,  {182)  ;  Not  always  expedient 
{183)  ;    When  either  m  or  p  is  a  positive  integer  {184)  ;    Examples . .   124-127 

Ibeational  Fbactions. — ^With  none  but  monomial  surds  {185)  ;    No  surd 

m 

except  of  the  form  (a  4-  6a;)"  {186)  ;    How  rationalized  when  the  surd  is 

of  the  form  Va-\-hx  ±l  xP-  {187)  ;    Examples 127-129 


SECTION  IK 

INTEGRATION  BY  PARTS. 

FoEMULA  POE  Integbation  BY  Pabts  {188) 130 

FoBMUL^  OF  Reduction.— Formula  ^  {189) ;  Formula  ^  {190)  ;  Form- 
ula <g  {191)  ;  Formula  ^  {192)  ;  Sometimes  fail,  Sch.  ;  Ex- 
amples   130-134 


CONTENTS.  XXV 

PAGE 

LoGABTTHMio  DiTPEEENTiAiiS. — Of  the  form  dy  =  X'  log^xdx  (193) ;  Ex- 
amples    134  135 

Exponential  Ditfebentiaxs. — Of  the  form  dy  =  x'*a'^dx  {194:) ;  Ex- 
amples   135,  136 

Special  forms  of  Expo>!entiax,3  136,  137 

Teigonometrical  DiFFEBENTiALS.  —Of  the  forms  dy  =  sm'»a;  dx,  dy  =  cos''x  dx, 
dy  =  sin'"  x  cos"  a;  d.v  {195) ;  When  this  process  is  appHcable  and  the  final 
forms,  {196) ;    Special  method  when  the  exponents  are  even  (page  140) ; 

Of  the  form  dy  =  ^^^^dx,  Ex.  17  ;    Examples 137-140 

^       cos»a; 

Of  the  forms  dy  =  x"  sin  «  dx  and  dy  =  x"  cos  x  dx  {197) ;    Examples 141 

p    Of  the  form  dy  =  sin"»  x  cos*  x  dx  integrated  in  terms  of  multiple  arcs  {198) ; 

Examples    141,  142 

CiECOLAB  DiffeeentiaijS. — Of  the  forms  dy  =f{x)  sin— ^  x  dx,  f{x)  cos— i  x  dx, 

etc.  {199)  :     Examples 142,  143 

Of  the  Foems  dy  =  e'^  sinj'xdx,  dy  =  e«3=cos"xdx  {200)  ;    Examples  . .  143,  144 

dy 
Op  THE  FoEM  di/ =  T — r-T r{201) 144,  145 

^       (a -f- fecosa;)"  ^ 


SECTION  K 

INTEGRATION  BY  INFINITE  SERIES. 

Occasion  foe  this  Method  {202) 146 

Examples  illustrating  the  method 146,  147 


SECTION  VL 


SUCCESSIVE  INTEGRATION. 


Geneeal  Peoblem  {203)  ;     Examples ;     The  constants    strictly  general, 

8ch.  1 ;    Condition  of  complete  integration,  8ch.  2 147,  148 

The  nth  Integbal  Developed  by  Maclaurin's  Formula  {204) ;    Example  . . .  149 


SECTION  VIL 

DEFINITE  INTEGRATION  AND  THE  CONSTANTS  OF  INTEGRATION. 

Definitions. — An  Indefinite  Integral  {205) ;  A  Corrected  Integral  (206) ; 
Integration  between  Limits  {207)  l  A  Definite  Integral  {208) ;  Ex- 
amples  150,  151 

Disposing  op  the  Constant  op  Integbation. — Two  methods  {209) ;  Ex- 
amples   151,  152 


INTRODUCTION. 


1  BRIEF  SURVEY  OF  THE  OBJECT  OF  PURE  MATHEMATICS 
AND  OF  THE  SEVERAL  BRANCHES. 

1,  JPiire  3fathe7natics  is  a  general  term  applied  to  several 
branches  of  science,  which  have  for  their  object  the  inyestigation 
of  the  properties  and  relations  of  quantity — comprehending  num- 
ber, and  magnitude  as  the  result  of  extension — and  of  form. 

2,  The  Several  branches  of  Pure  Mathematics  are  Arith- 
metic, Algebra,  Calculus,  and  Geometry. 

3,  Arithmetic,  Algebra,  and  Calculus  treat  of  number ;  and 
Geometry  treats  of  form  and  magnitude  as  the  result  of  extension. 

4,  Quantity  is  the  amount  or  extent  of  that  which  may  be 
measured  ;  it  comprehends  number  and  magnitude. 

The  term  quantity  is  also  conventionally  applied  to  symbols  used 
to  represent  quantity.  Thus  25,  m,  xi,  etc.,  are  called  quantities, 
although,  strictly  speaking,  they  are  only  representatives  of  quan- 
tities. 

ScH.  1. — It  is  not  easy  to  give  a  philosophical  account  of  the  idea  or  ideas 
represented  by  the  word  Quantity  as  used  in  Mathematics  ;  and,  doubtless, 
different  persons  use  the  word  in  somewhat  different  senses.  It  is  obviously 
incorrect  to  say  that  "Quantity  is  anything  which  can  be  measured." 
Quantity  may  be  affirmed  of  any  such  concept ;  nevertheless,  it  is  not  the 
thing  itself,  but  rather  the  amount  or  extent  of  it.  Thus,  a  load  of  wood,  or 
a  piece  of  ground,  can  be  measured  ;  but  no  one  would  think  of  the  wood 
or  the  ground  as  being  the  quantity.  The  qicantiiy  (of  wood  or  ground)  is 
rather,  the  amount  or  extent  of  it.  The  word  is  very  convenient  as  a  general 
term  for  mathematical  concepts,  when  we  wish  to  speak  of  them  without 
indicating  whether  it  is  number  or  magnitude  that  is  meant.  Thus  we  say, 
*'i7i  represents  a  certain  quantity,"  and  do  not  care  to  be  more  specific. 

As  applied  to  number,  perhaps  the  term  conveys  the  idea  of  the  whole, 
rather  than  of  that  whole  as  made  up  of  parts.  It  is,  therefore,  scarcely 
proper  to  speak  of  multiplying  by  a  quantity ;  we  should  say,  by  a  number. 


2  INTRODUCTION. 

On  the  other  hand,  when  we  apply  the  term  quantity  to  magnitude,  it  is  -with 
the  idea  that  magnitude  may  be  measured,  and  thus  expressed  in  number. 

The  distinction  between  quantity  and  number  is  marked  by  the  questions, 
"  How  much  ?"  and  "  How  many  ?" 

ScH.  2. — So,  also,  the  word  Magnitude,  as  used  in 'mathematics,  is  not 
easily  defined.  Sometimes  it  has  reference  to  quantity  in  the  aggregate,  or 
mass,  and  sometimes  to  the  relation  which  one  quantity  bears  to  another. 
Thus,  we  speak  of  a  line,  a  surface,  or  a  solid,  as  a  magnitude,  simply  mean-  • 
ing  thereby  that  these  have  extent, — are  extended.  A  circle,  a  triangle,  a 
cube,  are  magnitudes, — i.  e.,  they  have  extension.  Again,  we  speak  of  the 
magnitude  of  a  circle,  meaning  its  size, — area  as  compared  with  some  other 
surface.  The  magnitude  of  a  line  is  expressed  by  telling  how  many  times 
it  contains  another  Hne  of  known  length.  In  like  manner  the  magnitude 
of  a  surface  or  a  volume  is  made  known  by  comparing  the  surface  with  some 
unit  of  surface,  and  the  volume  with  some  unit  of  volume.  In  one  aspect, 
therefore,  number  is  an  expression  for  the  ratio  of  magnitudes. 

5,  WuiTlbev  is  quantity  conceived  as  made  up  of  parts,  and 
answers  to  the  question,  "How  many?' 

IiiiiUSTRATioN. — Thus,  a  distance  is  a  quantity  ;  but  if  "we  call  that  distance  5,  we 
convert  the  notion  into  number,  by  indicating  that  the  distance  under  consid- 
eration is  made  up  of  parts.  Now,  the  distance  may  be  just  the  same,  whether 
we  consider  it  as  a  whole,  or  think  of  it  as  5, — L  e..  as  made  up  of  5  equal  parts. 
Again,  m  may  mean  a  value,  as  of  a  farm.  We  may  or  may  not  conceive  it  as  a 
number  (as  of  dollars).  If  we  think  of  it  simply  in  the  aggregate,  as  the  worth 
of  a  farm,  m  represents  quantity  ;  but  if  we  think  of  it  as  made  up  of  parts  (sis 
of  dollars),  it  is  a  number. 

6.  Number  is  of  two  kinds.  Discontinuous  and  Contin- 
uous. 

7.  Discontinuous  JS^umber  is  number  conceived  as  made  up 
of  finite  parts  ;  or  it  is  number  which  passes  from  one  state  of  value 
to  another  by  the  successive  additions  or  subtractions  of  finite  units, 
— i.  e.,  units  of  appreciable  magnitude. 

8,  Continuous  Number^  is  number  which  is  conceived  as 

composed  of  infinitesimal  parts ;  or  ib  is  number  which  passes  from 
one  state  of  value  to  another  by  passing  through  all  intermediate 
values,  or  states. 

III. — The  method  of  conceiving  number  with  which  the  pupil  has  become 
familiar  in  arithmetic  and  algebra,  characterizes  discontinuous  number.  Thus 
the  number  13  is  conceived  as  produced  from  5  by  the  successive  additions  of 
finite  units,  either  integral  or  fractional.  In  either  case  we  advance  by  succes- 
sive steps  of  finite  length.  If  we  say  5,  G,  7,  etc.,  till  we  reach  13,  we  pass  by 
one  kind  of  steps  ;   and,   if  we  say  5.1,  5.2,  5.3,  etc.,   till  we  reach  13,   we  pass 


OBJECT  OF  PURE  MATHEMATICS. 


B 


Fig.  3. 


by  another  sort  of  steps  (tenths),  hut  as  really  hy  finite 
ones.  If,  however,  we  call  the  line  A  B,  Mg.  1,  x, 
and  CD,  x',  and  conceive  AB  to  slide  to  the  po- 
sition C  D,  increasing  in  length  as  it  moTes  so  as  to 
keep  its  extremities  in  the  lines  OM  and  ON, 
it  will  pass  by  infinitesimal  elements  of  growth  from  ^ 

the  value  x,  to  the  value    rj';   or,  it  Vviil  pass  from 

one  value  to  the  other  by  passing  through  all  inter-  ^i^-  !• 

mediate  values,  and  thus  becomes  an  illustration  of  continuous  number. 

Again,  if  the  line  AB,  Fig.  2,  be  considered  as 
generated  by  a  point  moving  from  A  to  B,  and  vro  AC  B 

call  the  portion  generated  when  the  point  has  reached  Fig.  2. 

C,  X,  and  the  whole  hne  x',  x  will  pass  to  x'  by  re- 
ceiving infinitesimal  increments,   or  by  passing  through  aU  states  of  value  be- 
tween x  and  x\ 

A  surface  may  be  considered  as  generated  by  the  motion  of  a  hne,   and  thus 

afford  another  illustration  of  continuous  number. 
Thus  let  the  parallelogram    AF  be   conceived  as 

generated  by  the  right  lino  A  B  moving  parallel  to 

itself  from  A  B  to  E  F.     When  A  B  has  reached 

the  position  CD,  call  the  surface  traced,   namely 
A  BCD,  X,  and  the    entire  surface    ABEF,   x'; 

then   will   x   pass  to   x'   by   receiving  infinitesimal  increments^  or  by  passing 

through  all  intermediate  values. 

Finally,  as  volumes  may  be  conceived  as  generated  by  the  motion  of  planes, 

all  geometrical  magnitudes  afford  illustrations  of  continuous  number. 

"We  usually  conceive  of  time  as  discontinuous  number,  as  when  we  think  of 

it  as  made  up  of  hours,  days,  weeks,  etc.     But  it  is  easy  to  see  that  such  is  not 

the  way  in  which  time  actually  grows.     A  period  of  one  day  does  not  grow  to 

be  a  period  of  one  week  by  taking  on  a  whole  day  at  a  time,  or  a  whole  hour, 

or  even   a  whole    second.     It  grows    by  imperceptible    increments    (additions). 

These  inconceivably  small   parts  of  which  continuous    number  is  made  up  are 

called  I:ifinitcsim,als. 

Motion    and    force  afford  other  illustrations  of   continuous  number.     In  fact, 

the  conception   which    regards   number    as  continuous,  will  be  seen  to  be  less 

artificial — more  true  to  nature — than  the  conception  of  it  as  discontinuous. 

9.  A.rithmetic  treats  of  Discontinuous  Number,  of  its  nature 
and  properties,  of  the  various  methods  of  combining  and  resolving 
it,  and  of  its  appHcation  to  practical  affairs. 

For  an  outUne  of  the  topics  of  Arithmetic,  see  the  Complete  School  Algebra, 
Akt.  9. 

10.  Algebra  treats  of  the  Equation,  and  is  chiefly  occupied 
in  explaining  its  nature,  and  the  methods  of  transforming  and 
reducing  it,  and  in  exhibiting  the  manner  of  using  it  as  an  instru- 
ment for  mfirhf^matioal  investiiration. 


4  INTRODUCTION. 

For  a  fall  account  of  the  province  of  Algebra,  see  the  Complete  School  Algebra, 
Aet.  10. 

H,  Calculus  (The  Infinitesimal  Calculus)  treats  of  Continu- 
ous Number,  and  is  chiefly  occupied  in  deducing  the  relations  of 
the  infinitesimal  elements  of  such  number  from  given  relations  be- 
tween finite  values,  and  the  converse  process,  and  also  in  pointing 
out  the  nature  of  such  infinitesimals  and  the  methods  of  using 
them  in  mathematical  investigation. 

12*  (jreOTyietvy  treats  of  magnitude  and  form  as  the  result  of 
extension  and  position. 

ScH.  1. — The  principal  divisions  of  the  science  of  Geometry  are  : 

1.  The  AjQcient,  Platonic,  Special,  Graphic,  or  Direct  Geometry  (the 
common  Geometry  of  our  schools),  iucludiug  Trigonoruetry,  Conic  Sec- 
tions, and  all  other  geometrical  inquiries  conducted  upon  these  methods. 

2.  The  Analytical,  Modern,  Cartesian,  General,  or  Indirect  Geometry 
i^the  theme  of  this  volume),  and 

3.  Descriptive  Geometry. 

ScH.  2. — The  first  system  of  geometrical  investigation  probably  took  its 
rise  as  a  science  in  the  school  of  Plato  (about  400  B.  C),  and  was  brought 
almost  to  its  present  state  of  perfection,  as  far  as  its  methods  are  concerned, 
by  the  time  of  Euchd  (about  300  B.  C);  hence  it  is  called  the  Ancient  or 
Platonic  Geometry.  As  the  argument  is  carried  forward  by  a  direct  inspec- 
tion of  the  forms  (figures)  themselves,  delineated  before  the  eye,  or  held  in 
the  imagination,  it  is  called  the  Direct  or  Graphic  method.  Inasmuch 
as  it  discusses  particular  instead  of  general  problems  it  is  properly  charac- 
terized as  Special.  With  this  method  of  geometry  the  student  is  supposed 
to  be  acquainted  before  commencing  the  study  of  this  volume. 

The  fundamental  notion  of  the  Modem  Geometry  (a  system  of  co- 
ordinates), was  developed  by  Des  Cartes  in  the  earher  part  of  the  17tli 
century,  and  hence  the  names  Modern  or  Cartesian.  The  term  Analytical 
has  come  to  be  applied  in  mathematics  in  the  sense  of  Algebraical,  all 
investigations  carried  forward  chiefly  by  the  aid  of  Algebra  being  called 
Arjalytical.  This  use  of  the  term  is  quite  unfortunate,  inasmuch  as  the 
processes  of  Algebra  are  no  more  analytical,  in  the  true  sense  of  that 
term,  than  r.re  those  of  the  Special  Geometry.  Again,  as  a  name  for  the 
General  Geometry,  even  if  used  in  the  sense  of  algebraic,  the  term  does 
not  distinguish  the  system  from  any  other  api)lication  of  algebra  to 
I  geometry. 

The  true  character  of  the  Modern  Geometry  is  expressed  by  the  terms 
Indirect,  and  General.  This  system  of  geometrical  reasoning  proposes 
the  solution  of  general  problems,  and  effects  its  purpose  by  first  translat- 
ing geometrical  forms  into  equations,  then  carrying  forward  the  investi- 
gation by  means  of  these  equations,  and  finally  returning  to  the  geomet- 
rical forms  by  a  re-translation.     The  indirectness  of  this  method  is  appa- 


OBJECT  OF  PURE  MATHEMATICS.  5 

rent,  and  might  iseem,  in  itself,  a  serious  objection  ;  but  it  is  found  to 
be  of  great  advantage,  inasmuch  as  it  makes  the  discussions  much  more 
comprehensive  {general).  To  illustrate  this  general  (comprehensive)  char- 
acter of  its  discussions,  we  have  only  to  notice  some  of  its  problems.  Thus 
the  Special  Geometry  discusses  the  problem  of  the  tangent  to  a  circle, 
and,  on  an  independent  basis,  investigates  the  properties  of  a  tangent  to 
any  other  curve,  making  a  special  problem  with  respect  to  each  sei3arate 
curve  studied.  On  the  other  hand,  the  General  Geometry  proposes  the 
problem  in  this  way  :  To  find  a  formula  sufficiently  general  to  embrace  the 
2'>roperties  of  tangents  to  all  plane  curves  ; — in  technical  language.  To  find 
the  equation  of  the  tangent  to  any  plane  curve.  Again,  in  the  Special  Ge- 
ometry, the  area  of  a  circle  is  obtained  (approximately).  But  the  General 
Geometry  proposes  to  investigate  the  problem  on  a  broader  basis,  and  find 
a  formula  which  shall  be  applicable  in  finding  the  area  of  any  plane  curve. 

IS,  DescTiptive  Geometry  is  that  system  of  geometry 
which  seeks  the  graphic  solution  of  geometrical  problems  by  means 
of  projections  upon  auxiliary  planes. 

This  is  the  ordinary  definition  of  the  Descriptive  Geometry,  and  it  would  be 
out  of  place  to  attempt  any  elucidation  of  it  here. 

ScH. — ^From  the  definition  of  Geometry,  as  well  as  from  the  detailed 
study  of  its  propositions,  it  will  be  seen  to  embrace  two  classes  of  prob- 
lems; viz.,  Problems  relating  to  Position,  and  Problems  relati7ig  to  Magni- 
tude. Problems  of  the  latter  class  were  solved  by  the  aid  of  algebra 
before  the  time  of  Des  Cartes  ;  but  it  was  reserved  for  him  to  invent  a 
method  by  which  problems  of  both  kinds  could  be  so  discussed.  This 
system  constitutes  the  foundation  of  the  General  Geometry. 

14:,  The  inquiries  in  the  General  Geometry  may  be  divided  into 

two  classes,  viz. : 

1.  Concerning  Plane  Loci, 

2.  Concerning  Loci  in  Space. 

In  accordance  with  this  division  the  present  treatise  is  divided 
into  Two  Books.^ 

ScH. — This  division  is  found  especially  convenient  when  the  subject 
is  treated  by  the  aid  of  the  Calculus,  as  it  corresponds  to  the  distinction 
between  functions  of  a  single  variable,  and  functions  of  two  variables. 


*  The  Second  Book  is  reserved  for  another  volume,  which  will  also  contain  an  advanced  course 
in  the  Calculus. 


BOOK  I. 

OF    PLANE    LOCI 


OHAPTEE  1. 

THE  CARTJESIAK  METHOD  OF  CO-OHDIHATES. 


snoTiojsr  I. 

Definitions  and  Fundamental  Notions. 

1,  The  term  Locus  as  used  in  geometry  is  nearly  synonymous 
with  geometrical  figure,  yet  having  a  latitude  in  its  use  which  the 
latter  term  does  not  possess.  The  locus  of  a  point  is  the  line 
(geometrical  figure)  generated  by  the  motion  of  the  point  accord- 
ing to  some  given  law.  In  the  same  manner,  a  surface  is  conceived 
as  the  locus  of  a  line  moving  in  some  determinate  manner. 

2.  TJie  General  QeoTnetry  is  a  system  of  geometrical  in- 
vestigation in  which  the  loci  under  consideration  are  represented  by 
equations,  and  the  inquiries  carried  forward  by  means  of  these 
equations,  the  final  object  being  the  discussion  of  general  problems. 

[Note. — ^WMle  it  is  true  that  the  only  way  to  obtain  a  full  comprehension  of  the  nature  of  a 
science  is  by  the  detailed  study  of  its  parts,  it  is,  nevertheless,  important,  at  the  outset,  to  com- 
prehend as  clearly  as  possible  the  general  aim  of  the  science,  in  order  that  the  tendency  of  the 
several  steps  in  our  progress  may  be  perceived,  and  the  symmetry  and  unity  of  the  whole  may 
appear .  According  to  our  definition  it  will  be  our  first  purpose  to  exhibit  a  scheme  by  whieh 
points,  lines  straight  and  curved,  the  magnitude  of  angles,  surfaces,  etc.,  which  we  have  char- 
acterized as  "  geometrical  forms  "  (loci),  may  be  represented  by  equations.  This  will  be  done  in 
Section  1st  of  this  chapter.  Section  2nd  will  then  exhibit  a  method  of  constructing  the  geometrical 
figure  represented  by  any  given  equation.  Then  will  follow  a  series  of  sections  showing  how  the 
equations  of  loci  are  derived  from  the  definitions  of  the  figures.  This  series  of  sections  comprises 
•what  may  be  termed  the  translation  of  geometrical  forms  into  algebraic  equations,  and  wiU  answer 
such  questions  as  :  "What  equations  represent  points?  What  straight  lines?  What  circles? 
What  ellipses  ?  etc.,  etc."  Section  2nd,  which  shows  how  equations  are  translated  into  geometrical 
forms,  might,  perhaps,  with  strict  logical  propriety,  follow  instead  of  precede  this  series  of  sections  ; 
but  it  is  thought  the  present  arrangement  will  promote  clearness  of  conception.  The  first  three 
chapters  will  be  seen  to  be  preparatory.    It  is  not  their  purpose  to  develop  geometrical  truths,  but 


DEFINITIONS  AND   FUNDxVMENTAL  NOTIONS.  7 

simply  to  prepare  instruments  (the  equations  of  loci)  to  be  subsequently  used  in  conducting  geo. 
metrical  inquiries.  In  the  fourth  chapter  it  will  be  our  purpose  to  show  how  geometrical  truth 
can  be  developed  by  means  of  these  equations.] 

3,  A  device  by  means  of  which  we  are  enabled  to  represent  loci 
by  equations  is  called  a 

METHOD  OF  CO-ORDINATES. 

4,  There  are  two  systems  of  co-ordinates  in  common  use,  viz.: 

1.  The  system  of  BectiHnear  Co-ordinates, 

2.  The  system  of  Polar  Co-ordinates. 

Sm  There  are  two  varieties  of  the  rectihnear  system  of  co-ordi- 
nates, the  rectangular  and  the  oblique.  (In  our  study,  the  rectan- 
gular system  will  always  be  used  unless  otherwise  specified. ) 

0,  In  order  to  locate  a  point  in  a  plane  by  the  method  of  recti- 
linear co-ordinates,  two  lines  intersecting  each  other  are  assumed 
as  fixed  in  position.  These  hues  are  called  A.xes  of  JtefevefiCCf 
or,  simply.  The  A.xes.  The  system  is  called  rectangular  or 
obhque,  according  as  these  lines  make  a  right  or  an  oblique  angle 
with  each  other. 

7,  One  of  these  axes  is  called  the  Axis  of  Abscissas^  and 
the  other  is  called  the  Axis  of  Ordinates, 

8,  The  Ori(/ifl  is  the  intersection  of  the  axes. 

9,  TJie  Co-ordinates  of  a  point  are  its  distances  from  the 
axes,  the  distance  to  either  axis  being  measured  on  a  line  parallel 
to  the  other,  or  on  that  other  axis. 

10»  The  Abscissa  of  a  point  is  the  co-ordinate  which  is 
measured  parallel  to  or  on  the  axis  of  abscissas,  and  is  the  distance 
of  the  point  from  the  axis  of  ordinates  measured  on  a  line  parallel 
to  the  axis  of  abscissas. 

11,  The  Ordinate  of  a  point  is  the  co-ordinate  which  is 
measured  parallel  to  or  on  the  axis  of  ordinates,  and  is  the  distance 
of  the  point  from  the  axis  of  abscissas  measured  on  a  line  parallel 
to  the  axis  of  ordinates. 

ScH.  1. — These  lines,  when  spoken  of  separately,  should  be  distinguished 
as  abscissa  and  ordinate;  but,  when  taken  together,  they  are  called  co- 
ordinates. 


8  THE   CARTESIAN   METHOD   OF   CO-ORDINATES. 

III. — Definitions    3    to    11    may  be  illuB-  Y 

trated  thus  :    Let  the  plane  in  which  the  loci  r/  » 

are  ^tuated  be   represented    by   the    surface  p'        /  ,  / 

of   the  paper,  Fig.  4.     In  this  plane  assume  /        /  / 

two    fixed,    indefinitely      extended,    straight      D"    /        / D"'    / 

I'nes,   as    XX'  and  YY,  intersecting  each  /  ^^j  /      D      X 

other  at    A,  and  to  which  all  points  in  the  /  /  P'" 

plane  are  to  be   referred.       These  lines  are  p''"  l'^" 

the  Axes   and  A  is  the  origin,  i.e.,  the  point  /, 

at   which    the  co-ordinates  are    conceived   to 
originate,  and    from    which    they    are    reck-  ^^' 

oned.  One  of  these  lines,  as  XX'  (in  ordinary  use  the  horizontal  one)  is  called 
the  Axis  of  Abscissas,  because  abscissas  are  reckoned  on  it  ;  and  the  other, 
YY',  is  for  a  hke  reason  called  the  Axis  of  Ordinates.  The  system  is  called 
rectangular  or  obUque  according  as  Y  AX  is  a  right  or  an  oblique  angle.  It  is 
evident  that  we  can  now  define  the  position  of  any  point  in  this  plane  by  giving 
its  distances  from  these  two  fixed  lines,  or  axes.  For  convenience,  we  measure 
these  distances  on  hues  parallel  to  the  axes.  (In  the  case  of  rectangular  axes, 
the  co-ordinates  will  become  the  perpendicular  distances  of  points  from  the 
axes.)  Thus  the  location  of  the  point  P  is  determined  by  giving  the  lengths 
of  PE,  the  abscissa  of  P,  and  of  PD,  the  ordinate  of  the  point.  Usually, 
A  D  is  called  the  abscissa,  instead  of  P  E.  P  D  and  A  D  taken  together  are 
called  the  co-ordinates  of  the  point  P. 

.  ScH.  2. — The  pupil  will  see  that  this  device  for  locating  points  is  not 
unlike  the  method  of  locating  places  on  the  earth's  surface  by  means  of 
latitude  and  longitude. 

12,  Abscissas  are  represented  in  the  notation  by  the  letter  ar, 
and  ordinates  by  y. 

13.  The    four    angles  into  which  the  plane    is  divided  hy  the 

axes  are  distinguished  thus  :  The  angle  above  the  axis  of  abscissas 
and  at  the  right  of  the  axis  of  ordinates  is  called  the  First  Angle; 
and  the  numbering  proceeds  from  right  to  left.  YAX  is  the  First 
Angle,  VAX'  is  the  Second,  X'AY'  is  the  Third,  and  XAY'  is  the 
Fourth. 

14:,  In  order  to  indicate  in  which  of  the  four  angles  a  point  is 
located,  the  signs  -f  and  —  are  used  on  the  following  principles  : 
abscissas  reckoned  from  the  origin  to  the  right  are  marked  4-,  and 
those  reckoned  to  the  left  are  — • ;  ordinates  reckoned  upward  from 
the  axis  of  abscissas  are  +,  and  those  reckoned  downwards  are  — . 
Accordingly,  the  abscissas  of  points  in  the  1st  and  4th  angles,  as  A  D 
and  AD'"  arc  +,  while  those  in  the  2nd  and  3rd  angles  as,  AD'  and 
AD",  arc  — .  Ordinates  in  the  1st  and  2nd  angles,  as  PD  and  P'D', 
are  4-^  and  those  in  the   3rd  and  4th,  ar;  P"D"aiid  P"'D"'  are — . 


DEFINITIONS  AND  FUNDAMENTAL  NOTIONS.  9 

IS*  The  quantities  used  in  General  Geometry  are  distinguished 
as  Constant  and  Variable. 

16,  A.  coiistaflt  quantity  is  one  which  maintains  the  same 
value  throughout  the  same  discussion,  and  is  represented  in  the  no- 
tation by  one  of  the  leading  letters  of  the  alphabet. 

17 »  Variable  quantities  are  such  as  may  assume  in  the  same 
discussion  any  value,  within  certain  limits  determined  by  the  na- 
ture of  the  problem,*  and  are  represented  by  the  final  letters  of 
the  alphabet. 

III. — In  Fig.  5  let  BECF   be  a  circle  whose  radius  is 
B,   and  XX'  and   YY'  be  the  axes  of  reference.    Eepre- 
Bent  the  abscissa  of  any  point  in  this  circumference  by  x 
and  the  corresponding  ordinate  by  y ;  bo  that  when  x 
signifies  AD,  y  shall  represent  PD  ;    when  — xis 
AD',  ?/ shall  be  P'D'  ;   when  — a;  is  AD",  — j/ shall 
be   P"D",  etc.     Now,   suppose   it   possible   to  represent 
the  relation  between  cc,   y  and  R  by  an  equation  so  gen- 
eral as  to  be  true  for  all  points  in  this  circumference,  as 
P,  P',  P",   etc.     (It  will  subsequently   appear  that  this 
equation  is  x-  -f-  ?/■  =  R' •)     lu  such  an  equation  B  would 

be  constant,  for  it  remains  the  same  for  all  positions  of  the  point  P  ;  and  x  and  y 
would  be  variables,  since  they  vary  in  value  with  every  change  of  the  position  of 
P.  In  such  a  problem  it  is  evident  that  x  ox  y  could  not  exceed  B,  hence  these 
variables  could  have  all  values- between  the  limits  of  -{-  B  and  —  B. 

Sen.  1. — Care  should  be  taken  not  to  confound  the  terms  co7is^«wif  and 
variable  as  here  used,  with  known  and  unknown  as  used  in  aJgebra  : 
esi^ecially  as  the  notation  would  suggest  an  identity  which  does  not  exist. 
Botli  the  known  and  unknown  quantities  of  Algebra  are  constants ;  moreover 
the  constants  in  General  Geometry  may  be  either  knoimi  or  unknown;  and 
the  same,  in  a  certain  sense,  may  be  said  of  the  variables. 

ScH.  2. — In  order  that  the  variables  may  retain  their  peculiar  charac- 
teristic, we  cannot  have  as  many  equations  arising  from  a  particular  prob- 
lem as  there  are  variables  ;  thus,  if  there  are  two  variables  involved,  we  have 
but  one  equation.  In  algebra  such  problems  are  called  indeterminate,  since 
the  equation  does  not  determine  definite  values  of  the  unknown  quantities, 
but  can  be  satisfied  by  an  infinite  variety  of  values .  From  this  feature  of 
the  General  Geometry  it  is  sometimes  called  Indeterminate  Analysis.  The 
Calculus  is  also  embraced  under  the  same  term,  as  its  problems  involve  a 
like  feature. 


*  Our  limits  do  not  permit  a  discussion  of  the  continuity  of  functions  and  the  general  geo- 
metrical interpretation  of  imaginary  co-ordinates,  and  hence  for  simplicity  we  retain  the  con- 
ception of  imaginaries  as  impossible  quantities. 


10 


THE  CARTESIAN  METHOD  OP  CO-OBDINATES. 


IS.  To  construct  an  equation,  or  find  its  locus,  is  to  draw  the  g&^ 

metrical  figure  represented  by  it. 


■^♦»- 


SECTION  IL 
Constructing  Equations,  or  Finding  their  Loci. 

19.  A  curve  is  continuous  when  its  course  is  uninterrupted 

both  in  extent  and  in  the  character  of  its  curvature. 


III. — A  circle,  an  ellipse,  and  the  curve  in  Mg.  6  are 
examples  of  curves  continuous  in  extent  and  curvature. 
They  may  be  traced  throughout  by  the  uninterrupted 
movement  of  a  point.  The  curve  Fig.  1,  is  discontinuous 
in  extent ;  and  in  Fig.  S,  we  have  an  example  of  a  curve 
discontinuous  in  curvature.  Fig .  9  affords  an  example  of 
discontinuity  both  in  extent  and  curvature. 

20,  A.  ^Branch  is  a  continuous  portion  of 
a  curve.  In  Figs.  7  and  8  the  curves  have  two 
branches  each.  In  Fig.  9  there  are  four 
branches. 

21.  A  curve  is  symmetrical  with  respect 
to  either  axis,  or  to  any  line,  when  it  has 
the  same  form  on  both  sides  of  the  hne,  or 
when  every  point  on  one  side  of  the  line  has  a 
corresponding  point  on  the  other.  The  curves 
in  Figs.  6,  7  and  9  are  symmetrical  with  re- 
spect to  the  axis  of  abscissas,  and  the  first  two 
with  respect  to  both  axes.  The  curve  in  Fig.  8 
is  not  S3^mmetrical  with  respect  to  any  hne. 


Fig.  6. 


Fig.  7. 


Fig.  8 


Fig.  9. 


22 


JProh.   To  locate  a  Point  whose  co-ordinates  are  given. 


Solution.  — Lay  off  from  the  origin,  on  the  axis  of  abscissas,  a  distance  equal  to 
the  given  abscissa,  to  the  right  if  the  abscissa  is  -}-,  and  to  the  left  if  it  is  — . 
Through  the  point  thus  found  draw  a  line  parallel  to  the  axis  of  ordinates,  and 
lay  off  on  it  a  distance  from  the  axis  of  abscissas  equal  to  the  given  ordinate,  above 
if  the  ordinate  is  -|->  and  below  if  it  is  — .  The  point  thus  found  wUl  be  the  one 
required. 


CONSTRUCTING  EQUATIONS.  H 


Ex.  1.  Locate  the  point  x  =  3,  y=  —  5. 


Solution. — ^Draw  the  axes  XX' and  YY'.     Lay      ^' 
off  A  B  =  3  to  the  right,  as  a*  is    -\-,  and  draw  BO 
parallel  to  YY'.     Then  take  PB  =5  helow  the  axis 
of  abscissas  as  y  is  — ,  and  P  is  the  point  required. 


Y' 


o 


ScH. — Points   are  usually  designated  by  mentioning  Fig.  10. 

simply  their  co-ordinates,  as  the  point  3,  — 5,  for  the 
point  in  the  last  example.     The  abscissa  is  mentioned  first. 

Exs.  2  to  9.  Locate  —6,  2;  —5,  —7;  —3,  0  ;  0,  — 3  ;  0,  0  ;  5,  —1; 
2,  0  ;  0,  4. 

Queries. — Where  are  points  situated  whose  abscissas  areO?  Where  are  points 
situated  whose  ordinates  are  0  ?  What  are  the  co-ordinates  of  the  origin  ?  In  what 
line  are  3,  — 2  ;  3,  5  ;  3,  0  ;  and  3,  4  situated  ? 


23,  JProb,  To  find  the  locus  of  an  equation  between  two  variables; 
i.   e.,  to  construct  the  equation. 

Solution. — Solve  the  equation  with  respect  to  one  of  the  variables.  Then, 
since  the  equation  expresses  the  relation  between  the  co-ordinates  of  all  points  in 
the  locus,  substitute  for  the  other  variable  any  values  which  give  real  values  for  the 
first,  and  locate  the  points  thus  determined.  These  will  be  points  in  the  locus  ; 
and,  by  determining  a  sufi&cient  number,  the  locus  can  be  sketched  through  them. 

Ex.  1.  Construct  the  equation  ^^ — - —  =  1. 

o 

Solution. — Solving  the  equation  for  y,  we  have  y  =  2x  -^  S.     Now  attributing 
arhitary  values  to  x,  we  make  the  following  table  of  corresponding  values  : 
When  X  =1,     y  =5,    giving  the  point  1,  5  ; 
"    x==2,     y=7,       "        "       "      2,  7  ; 
"    a;  =  3,     7/ =  9,       "         "        "      3,  9  ; 
etc.,         etc.,         "         "        '*       etc. 
Noticing  that  all  positive  values  of  x  give  real,  positive,  and  single  values  to  y,  we 
discover  that  the  locus  has  but  one  branch  which  extends  to  the  right   of  the 
axis  of  ordinates,  extends  indefinitely,  and  lies  above  the  axis  of  abscissas. 
Again,  giving  negative  values  to  a',  we  have 

When  X  =  — 1,     y  =  1,    giving  the  point  — 1,  1  ; 
"    X  =  —2,     y  =  —1,     ''       "       "      —2,  — 1  ; 
«'    ic  =  —3,     y  =  —3,     "      "       "      —3,  —3  ; 

and  for  all  subsequent  negative  values  of  x,  y  has  real,  negative,  and  single  values. 
Hence  we  learn  that  the  locus  has  a  single  branch  extending  indefinitely  in  the  third 
angle. 

If  we  make  2/  =  0,  a;  =  —  li  ;  whence  we  see  that  the  locus  cuts  the  axis  of  abscis- 
sas at  — li,  0.  If  we  make  x  =  0,  y  =  S  ;  and  hence  the  locus  cuts  the  axis  of 
ordinates  at  0,  3. 


12 


THE  CARTESIAN   METHOD   OF    CO -OBDINATES. 


Finally,  locating  these  points,  as  in  Fig.  11,  we  find  that 
the  line  M  N  includes  all  the  points,  and  hence  conclude 
that  it  is  the  required  locus. 

ScH. — If  any  other  values  be  attributed  to  x,  either 
integral  or  fractional,  positive  or  negative,   and 
the  corresponding  values  of  3/  deduced,  the  points 
thus  determined  will  fall  in  the  line  M  N. 


0/3 


X' 


A 


X 


Ex.  2.    Find  the  locus  of    the    equation 

Fig.  11. 

Result.  A  straight  line  cutting  the  axis  of  abscissas  at  8,  0,  and 

the  axis  of  ordinates  at  0,  2. 

^ /)/ 

Ex.  3.  Find  the  locus  of  the  equation  ^x  —  1  =     -—-. 


Result  A  right  line  passing  through  0,  4,  and  3,  —  1. 

24:.  ScH. — ^If  there  is  nothing  in  the  nature  of  the  equation  to  make  an- 
other course  preferable,  it  is  customary  to  solve  it  for  y,  finding  the  value  in 
terms  of  x,  and  constants.  If,  however,  the  equation  is  above  the  second 
degree  with  respect  to  either  of  the  variables,  it  is  expedient  to  solve  it  with 
reference  to  the  variable  which  is  least  involved.  Thus,  in  order  to  construct 
Zx  —  y-  =  2j/3  —  y  —  5,  we  solve  with  reference  to  x,  and  then  substitute 
arbitrary  values  for  y,  finding  the  corresponding  values  of  x. 

26.  Def. — The  Independent  Variable  is  the  one  to  which 
we  assign  arbitrary  values,  usually  x.  The  other  is  called  the 
Dependent  Variable. 

This  distinction  is  made  simply  for  convenience,  and  is  not  founded 
in  any  difference  in  the  nature  of  the  variables  :  either  variable  may 
be  treated  as  the  independent  variable. 

2(>,  ScH. — There  are  certain  peculiarities  of  loci,  which  readily  appear 
from  the  form  of  the  equation.  These  should  always  be  noted.  Observing 
tiiein  is  called  Discussing,  or  Interpreting  the  Equation.  The  following  are 
some  of  these  points  : 

1st.  The  Intersection  of  the  locus  loith  the  Axes.  Where  the  locus  cuts  the 
axis  of  abscissas  ?/  =  0;  hence  substituting  this  value  (0)  of  y,  in  the  equation, 
and  finding  the  corresponding  value  or  values  of  x,  determines  the  intersec- 
tions with  the  axis  of  abscissas.  In  like  manner,  making  x  =  0,  and  finding 
the  corresponding  values  of  y,  determines  the  intersections  with  the  axis  of 
ordinates. 

2nd.  The  Limits  between  which  the  locus  is  comprised,  and  its  continuity  or 
discontinuity  between  these  limits.  These  questions  are  to  be  determined  with 
respect  to  each  axis.  The  limits  are  discovered  by  determining  the  great- 
est and  least  values  of  the  independent  variable  which  give  rea/ values  to  the 


CONSTRUCTING  EQUATIONS.  13 

dependent  one.  If  all  values  of  the  independent  variable  between  the  Hmits 
observed  in  this  way,  give  real  values  for  the  dependent  variable,  the  locus 
is  continuous  in  extent  between  these  hmits.  If,  on  the  other  hand,  there 
are  certain  values  oi  the  former  which  render  the  latter  imaginary,  the  locus 
is  discontinuous;  and  the  limits  of  discontinuity  are  to  be  observed  by  find- 
in<^  the  limits  between  which  the  values  of  the  dependent  variable  are  ima- 
ginary. 

3rd.  Whether  the  locus  is  symmetrical  with  respect  to  an  axis,  or  with  any 
line,  or  not  The  manner  of  determining  this  is  as  follows  :  If,  for  each  real 
value  of  one  variable,  the  other  has  two  values,  numerically  equal  but  with 
contrary  signs,  there  are  points  similarly  situated  on  opposite  sides  of  the  axis 
from  which  the  variable  having  two  values  is  reckoned,  and  hence  the 
locus  is  symmetrical  with  respect  to  that  axis.  Again,  if  there  is  any  line  so 
situated  that  the  values  of  the  intercepts  of  either  of  the  co-ordinates  be- 
tween It  and  the  locus,  on  both  sides  of  the  line,  are  equal,  the  locus  is 
^symmetrical  with  respect  to  that  Hne. 

[Note. — There  are  mauy  other  characteristic  features  of  loci  which  appear  more  or  less  immedi- 
at^i^y  from  the  forixi  of  the  equation,  and  some  of  which  will  be  noticed  in  a  subsequent  part  ox  the 
course.  Those  now  mentioned  are  sufficient  for  our  present  purpose  if  the  pupil  becomes  perfectly 
familiar  with  them.  This  famiharity  can  be  attained  only  by  2  careful  study  of  examples.  In  fact  • 
it  is  hardly  probable  that  the  pupil  can  understand  the  full  purport  of  the  language  of  the  last 
scholium  until  he  has  solved  several  examples.  After  studying  a  few  which  follow,  he  can  retiim 
and  read  the  scholiiim  again,  and  be  better  able  to  see  its  meaning.] 

Ex.  4.  Find  the  locus  of  the  equation  x^  +  y'^  =  25. 


Solution.  ?/  =  ±  ^^25  —  a;2.  For  x  =  Q,  y  =  5  and  — 5.  Hence  the  locus 
cuts  the  axis  of  ordinates  at  (0,  5)  and  (0,  — 5).  For  ?/  =  0,  a;  =:  5  and  —  5. 
Heuce  the  locus  cuts  the  axis  of  abscissas  at  (5,  0)  and  ( — 5,  0).  Again,  as  every 
value  of  X  between  -f-  5  and  — 5,  gives  two  real  values  for  y,  numerically  equal,  but 
with  opposite  signs,  the  locus  is  symmetrical  with  respect  to  the  axis  of  abscissas, 

and  continuous  between  these  limits.     In  hke  manner,  cc  =  it  -n/25  ■ y~  shows 

that  the  locus  is  symmetrical  with  respect  to  the  axis  of  ordinates,  and  continuous 
between  y  =  5,  and  ?/  =^  —  5.  When  x  is  numerically  greater  than  5  (either  -f  or 
— ),  the  values  of  2/ bocome  iwagrmarj/.  Hence  the  locus  is  comprised  between  the 
limits  ic  =  5,  and  cc  =  —  5.  From  ic  =  dr  ^25  —  y',  it  appears,  in  like  man- 
ner, that  the  limits  in  the  direction  of  the  axis  of  ordinates  are  y  =  5,  and  —  5. 

Now  assigning  to  x  arbitrary  values  between  -f-  5  and  —  5,  we  find  the  following 
table  of  values,  and  points  in  the  locus  : 

When£e  =  l,  y=±z  v'24  =  ±:  4.9  nearly  ;  and  we  have  points  (1,  4.9)  and  (1, — 4.9); 

«'    £>;==2,  y.-^=fc  ^^21  ==h4.6  nearly;        "     "       "       "     (2,  4.6)  and  (2,— 4.6); 

*'    05=^3,  y=rfc^l6  =  =fc4  "     "      "       «'       (3,  4,)  and  (3, --4); 

**    a  =  4,  2/=.rt:v/9  =±3  "     "       ".      «      (4,  3,)  and  (4,— 3); 

For  negative  values  of  ;r  the  following  points  are  found  ( — 1,  4.9)  and  ( —  1, — 4.9); 
(—2,  4.6)  and  (—2,  —4.6);  (—3,  4)  and  (—3,  —4);  (—4,  3)  and  (—4,  —  3). 


ii 


THE   CARTESIAN   METHOD   OF  CO-OKDINATES. 


Constructing  the  points  thus  determined 
they  are  found  to  be  in  the  circumference  of  a 
circle  whose  radius  is  5,  and  which  is  sym- 
metrical with  the  axes,  as  in  Fig.  12.  It  is 
also  to  be  observed  that  any  values  of  x, 
fractional  as  well  as  integral,  between  the 
limits  x  =  5,  and  x  =  —  5,  give  values  for  y 
which  locate  points  in  the  same  circumference. 

Ex.  5.  Construct  and  discuss  the  equa- 
tion 9y2  -j-  4:X^  =  36. 

Solution. — Solving    the  equation   for  y,   y  = 
db  I'v/S' —  x2.  We  now  observe  that  for  ic  =  0,  ?/  := 

±  2 ;  therefore  the  locus  cuts  the  axis  of  ordinates  at  (0,  2)  and  (0, —  2).  In  like  manner, 
making  y  =  Q,x  =  zh^',  and  hence  the  locus  cuts  the  axis  of  abscissas  at  (3,  0)  and 
( —  3,  0).  Again,  for  each  value  of  x  which  renders  9  —  x-'^  0,  i.  e. ,  for  each  value  be- 
tween X  =  3,  and  —  3,  y  is  real  and  has  two  values,  numerically  equal,  but  with  con- 
trary signs  ;  therefore  the  locus  is  symmetrical  with  reference  to  the  axis  of  abscis- 
sas, and  continuous  between  the  hmits  cc  =  3,  and  x  =  —  3.  Beyond  these  values 
of  X,  y  becomes  imaginary,  and  the  locus  is  entirely  comprised  within  x  =  3  and 
X  =  — 3  along  the  axis  of  abscissas.  In  a  similar  manner  from  x  =  ±z  ^  *>/ ^  —  t/^, 
it  appears  that  the  locus  is  comprised  between  y  =  2  and  y  =  —  2,  and  is  sym- 
metrical and  continuous  with  respect  to  the  axis  of  ordinates. 

Finding  the  values  of  y  corresponding  to  a  sufficient  number  of  arbitrarily  taken 

values  of  x,  so  as  to  enable  me  to  sketch  the  curve,  we  have  the  following  table  of 

values  : 

For  a;  =  0,      ?/  =  =fc  2,  giving  the  points  a,  a'  in  J/  ig.   13  ; 

"         "  h,  V  "  " 

"         "  c   c'    '^  " 

"  d,  d'   "  ** 

cc          (c  ee'"  '* 

"    g,  g'  " 

"      h,  h'  "         " 

Since  the  equation  contains  only  the  square  of  x,  neg- 
ative values  of  x  give  the  same  values  for  y  as 
positive  values  do,  and  the  portion  of  the  curve 
on  the  left  of  the  axis  of  ordinates  is    sym- 
metrical with  that  on  the  right. 

Finally,    locating    the    points,    as    made 
known  in  the   table,    and   a  similar  set  of 
points  on  the  left  of  the  axis  of  ordinates,  we 
have  an  ellipse  whose  axes  are  6  and  4,  Fig.  13. 

Ex.  6.  What  is  the  locus  of  if-  =  2x  - 


For 

x 

= 

0, 

2/  = 

±  i 

i,  givi 

X 

= 

,5, 

y  = 

±: 

1.97, 

X 

= 

1, 

y  = 

db 

1.89, 

X 

=: 

1.5, 

y  = 

± 

1.73, 

,1" 

= 

2 

y  = 

± 

1.49, 

.1' 

= 

2.5, 

y  = 

:  ±2 

1.1 

a- 

= 

2.7£ 

\y== 

±: 

.8, 

X 

= 

2.9, 

y  = 

:  ±: 

.51, 

X 

= 

3, 

y  = 

=  0 

CONSTRUCTING   EQUATIONS. 


15 


Ans. — It  cuts  the  axis  of  abscissas  at 
3,  0,  and  lies  wholly  to  the  right  of  this 
point,  extending  indefinitely  in  two 
branches,  one  above  the  axis  of  abscissas 
and  one  below  it ;  and  the  two  are  sym- 
metrical with  this  axis.  Fig.  14.  The 
branch  B  M  extends  indefinitely  in  the 
1st  angle,  and  B  M '  in  the  fourth.  The 
locus  is  known  as  a  Parabola. 


Iy' 


Fig.  14. 


Exs.  7  to  10.    Construct  the  following  equations  :    ^x  -^  2?/  =  4 ; 
2x-\-3y=:0  ;  Sx^  -\-  5y^  =  12  ;  2/«  — ■  6?/  -f  ;y?  =  16. 

Ex.  11.  Find  the  locus  of  x^  —  y^  =  10. 
Ans. — The  locus  is  represented  in  Fig. 
15.     It  is  discontinuous  "between  x  =^ 

n/10  and  x  =  —  v^lO  ;  but  to  the  right 
and  left  of  these  points,  it  extends  indefi- 
nitely. It  is  symmetrical  with  respect 
to  both  axes.  The  curve  is  known  as 
an  Hyperbola. 


ly/ 

Pig.  15. 


Exs.  12  to  23.  Construct  and  discuss  the  following  :  y^~ 

16  —  j;2  .  y-i  =z  10^  —  ^2  ;  2/2  =  1207 ;  x^  —  6^  +  9  + 
2/2  +  10y==0;  25{y  +  4)^  +  16(^  —  5)^  =  400  ;  y^ 
=  4  +  2  (07  —  3)2  ;  2/2  =  072  —  4  ^  y2  =  Sx^  —  073  +  5  ; 
xy=16  ;  y^  =x^  —  074 ;  y'iz=x*  —  x^ ;  y^  =  x'*  —  x\ 

Ex.  24.  Construct  and  discuss  the  equation  x  =  log  y. 

Results. — Assuming  x  as  given  in  the  following  table  of 
values  (any  convenient  values  of  x  maybe  taken),  the 
values  of  y  can  be  found  from  a  table  of  logarithms. 


For  x=  0,  y  =  1. 
"  X  =  .2,  y  =  1.58  nearly. 
"  a;=  .4,  2/ =  2.51       " 
»  x  =  .e,  t/  =  3.98       " 
"  x  =  .8,  y=  6.31       " 
"  x=  1,  2/ =  10. 
etc.     etc.     etc.     etc. 


Forx= — .1,  2/ =•  8  nearly. 
"  a;  =  —  .22,  y—.6  " 
"  «  =  — .4,  y  ^.4  " 
"  x  =  —.7,  2/  =  -2  " 
"  x=—  1,  y  =  .l  '* 
"  ic  =  —  2, 2/  —  .01  '  * 
etc.     etc.    etc.     etc. 


X' 


Fig.  16. 

Locating  these  values,  we  have  the  curve   MN,  Fig.  16,  which  is  called  the 

Zogarithmic  Curve.     It  lies  wholly  above  the  axis  of  abscissas,  as  negative  numbers 

have  no  logarithms.     It  extends  on  both  sides  of  the  axis  of  ordinates,  and  cuts  it 

at  (0,  1, )  a  point  through  which  all  logarithmic  curves  pass,  in  whatever  system  the 


16 


THE  CARTESIAN  METHOD   OF   CO-ORDINATES. 


logarithms  be  taken,  since  log  1  =  0  in  all  systems.     The  curve  extends  inde- 
finitely to  the  right  and  to  the  left ;  but  the  portions  are  not  symmetrical. 

Ex.  25.  Construct  x  =  log  2/,  assuming  2  as  the  ha&e  of  the  system 
of  logarithms  ;  giving  2/  =  2"" . 

Tne  values  are,  a:=0,  y=\-^  x  =  l,  y=2;  x  =  2,  y  =  4,-^x=3, 
y  ==z  8;  etc.      Also,  x  =  — 1,  y=  .5  ;  x  ==  — 2,  y  =  .25  ;   x  = 


-3,  y 


a  5 


X  =  — 4,  y  = 


T¥>  ^'^^• 


QuEKiES. — Locating  this  curve  on  the  same  axes  with  the  preceding,  what  common 
point  do  they  possess  ?  Does  the  right  hand  branch  of  this  he  to  the  right,  or  to 
the  left  of  the  former  .'*  Does  the  left  hand  branch  approach  the  axis  of  abscissas 
more  rapidly,  or  less  rapidly  in  the  latter  than  in  the  former  ?  What  makes  these 
differences  ?     How  would  it  be  with  a  base  100  ? 

Ex.  26.   Construct  and  discuss  y  =  sin  x. 

SuGs.  — The  unit  arc  is  a  portion  of  the  circumference  equal  to  the  radius.     This 

arc  is  57.3°  nearly  ;  since,  radius  being  unity,  the  semi-circumference  is  3.1416,  and 

180^ 

Hence  the  following  table. 


3.1416  -""■"    ^^"^^•'• 

For  x=    0^  =     0, 

2/=     0 

"    £c  =  103  =  .17. 

2/ =  .17 

"    jc  =  20o  =  .35. 

2/ =  .34 

«    x  =  303  =  .52, 

2/ =  .50 

"    £C  =  403=.70, 

2/ =  .64 

"    £C  =  50^  =  .87, 

2/ =  .77 

etc.        etc.        etc. 

etc. 

This  curve  is  called  the  Sln- 
"usoid.  Where  does  it  cut  the 
axis  of  abscissas?  Is  it  hm- 
ited?  What  are  the  hmits  of 
y  ?  What  is  the  meaning  of  x 
=  —  10^,  x  =  —  200,  etc.  ? 


Exs.  27  to  33.  Construct  y  = 
tan  X  ;  y  ==  cot  x  ;  y  =  cos  x 
y  =  versin  x  ;  y  =  coversin^ 
y  z=  sec  X  ',  y  =  cosec  x. 


For  x  =  180°  =3.14, 
"  x  =  1900  =  3.31, 
"  a;  =  2000  =  3.49, 
"  a;  =  2100  =  3.66, 
"  ic  =  2200  =  3.84, 
"    a;  =  230O  =  4.01, 

etc.         etc.  etc. 

Y 


y  = 
y  = 
y  = 
y  = 


0 

.17 
.34 

.50 


2/  =  — .64 

y  =  -.ii 

etc. 


Y' 

Fig. 


17. 


ScH.  — These  loci  can  be  con- 
sfcracted  with  sufficient  accura- 
cy without  the  numerical  com- 
putations. Thus,  taking  the 
Ex.  y  =  tana;,  draw  a  circle  ON,  Fig. 
18,  with  any  convenient  radius.  Divide 
a  quadrant,  as  M  N,  into  equal  parts,  each 
so  small  that  for  practical  purposes  the 
chord  and  arc  may  be  considered  equal. 


7 

s 

5 

i 
3 

2 

i. 

1 

Y 

A/1 

1 

MX' 

/' 

■J 

/ 

1 

/  12  3d5( 

/ 

Y' 

>7       X 

Pig.  18. 


%. 


THE  POINT  IN  A  PLANE. 


17 


Estimating  the  tangents  and  arcs  from  M,  and  having  dra^vnthe  tangents  as 
in  the  figure,  lay  off  the  arcs  on  the  axis  of  abscissas.  At  the  extremity 
of  Al  lay  off  an  ordinate  equal  to  tangent  Mi,  etc.,  etc.  There  are  an 
infinite  number  of  similar  infinite  branches  to  this  curve. 

On  the  figure  used  for  getting  the  tangents,  when  the  arc  passes  90°  the 
tangents  (and  hence  the  brdinates)  become  negative.  Strictly  speaking, 
f  negative  values  of  x  would  be  obtained  by  measuring  the  arcs  on  the  circle 
from  M  downward,  or  from  left  to  right ;  so  that,  from  x=  0  to  .r  = 
—90'',  the  tangents  (and  hence  the  ordinates)  are  negative.  From  x  =  — 90° 
to  .^  =  —180°,  the  tangents  (and  hence  the  ordinates)  are  positive.  Where 
do  the  branches  cut  the  axis  of  abscissas  ?  At  what  values  of  x  do  the 
ordinates  become  infinite  ? 

^♦» . 


SUCTION  IIL 
The  Point  in  a  Plane. 

27*   Def. — The  Equations  of  a  I^oint   are  the  algebraic 

expressions  which  determine  its  position. 


28.  JProp,     The  Equations  of  a  Point  in  a  plane  are  x  =  a,  and 
y  =  b,  in  which  the  signs  of  a  and  b  are  general. 


p1 


B' 


Fr-pr 


D' 


Dem. — If,  as  in  Fig.  19,  we  make  AB  =  a,  and 
through  B  draw  DE  parallel  to  YY',  every  point 
in  DE  will  have  its  abscissa  equal  to  a.  In  like 
manner  make  AC  =  6,  and  draw  FG  parallel  to 
XX', and  every  point  in  FG  will  have  b  for  its 
ordinate.  Hence  the  point  P  has  a  for  its  abscissa, 
and  b  for  its  ordinate  ;  and  since  two  straight  lines 
can  meet  in  only  one  point,  P  is  the  onli/  point 
v/hich  has  these  co-ordinates.  Therefore  x=a,  and 
y  =  b,  determine  the  position  of  a  point,     q.  e.  d. 

ScH.  j^. — If  we  have  x  =  — a,  and  t/  =  b,  P'  is  the  point. 
3/  =  — b,  P"  is  the  point,  etc. 


|Y 


E' 


Df 


pTT-Qr 


Y' 

Fig.  19. 
li  x=  — a,  and 


ScH.  2. — If  a;  =  a  =  0,  and  y  =  h,  the  point  is  in  the  axis  of  ordinates. 
\i  X  =  a,  and  y  ==  5  =  0,  the  point  is  in  the  axis  of  abscissas,  x  =  0, 
2/  =  0  characterizes  the  origin. 

ScH.  3. — A  point  is  usually  designated  by  simply  naming  its  co-ordinates, 
'(h.Q  abscissa  being  mentioned  first.  Thus  the  point  (wi,  ti)  is  the  same  as 
the  point  x  =  m,  and  y  =  n. 

Exs.  1  to  6.      Locate  the  points  x  ==  — 3,  y  =  4 ;  (5,  — 7) ; 

(0,-5);  (0,4);  (0,0);  (6,0). 


18 


THE   CAETESIAN   METHOD   OP  CO-ORDINATES. 


Exs.  7  to  10.     How  are  the  points  (5,  %) ;  {%  —6) ;  (J,  m) ;  (— n,  J) 
situated  ? 

Answer  to  the  first. — In  a  line  parallel  to  the  axis  of  ordinates  and  at 
a  distance  5  from  it.  Any  point  in  this  line  fulfills  the  conditions, 
since  y  =  ^,  i.  e.,  is  indeterminate. 

Exs.  11,  12.  Construct  the  triangle  whose  yertices  are  ( — 3,  4) ; 
(5,  — 1);  and  (2,  — 6).  Also  the  triangle  whose  vertices  are  (0,  3) ; 
(—5,0);  and  (0,  0). 

Exs.  13,  14.  What  figure  is  that  the  vertices  of  whose  angles  are 
(2,  3);  (2,  8);  (7,  8);  and  (7,  3)?  What  figure  is  that  the  vertices  of 
whose  angles  are  (2,  9);  (—8,  9);  (—8,  1);  and  (2,  —1)? 


29.    JPvop,     The    Distance    between    two   points    in   a   plane   is 
sJX^'  —  x"Y-\-  {y'  —  y"y\  in  v)hich  {x',  y',)  and  {x'\  y")  arethe points. 


Dem. — Let  the  points  (a;',  y' )  and  (x"  y")be  represented 
by  P'  and  P",  as  in  Fig.  20,  and  tlie  distance  between 
them,  P'P",  by  D.  Draw  P"D  parallel  to  AX. 
Then  P"D  =  cc'  —  x",  and  P'  D  =  y'  —  y".  From 
the     right     angled     triangle     P'  P"  D,     we     have,     — 

D  =  '-/{x'  —  x"Y  -{-  {y'  —  y' )-.     Q.  E.  D. 


p7- 


B       C 

Fig.  20. 


CoK. — ^  either  of  the  points,  as  P'\  is  at  the  origin,  its  co-ordinates 
are  0,  0,  and  D  =  Vx'^  +  y'K 

QuEBiES. — When  P"  is  in  the  axis  of  abscissas  and  at  the  right  of  the  origin, 
what  is  the  formula?     The  same  with   P"  at  the  left  of  the  origin,  give  D  = 

^/(x'  +  x")'2 -f-y'^.  If  P'  is  in  the  1st  angle  and  P"  in  the  3rd,  what  is  the 
formula?  If  P'  is  in  the  axis  of  abscissas  and  P"  in  the  axis  of  ordinates?  If 
one  is  in  the  2nd  and  the  other  in  the  4th  angle  ? 


ScH. — Observe  that  the  formulaD  =  ^[x  —  x"Y  +  [y'  —  y")'^  is  strictly 
general,  only  noticing  carefully  the  effect  of  the  position  of  the  points,  upon 
the  signs  of  their  co-ordinates.  Thus  for  a  point  P",  in  the  4th  angle,  we 
have  x" ,  and  — y" ;  which,  substituted  in  the  formula,  gives  for  P'  in  the  1st 
angle  and  P"  in  the  4th,  D  — -  v  {x  —  x"Y-\-  {y'  -f-  y"  Y' 

Examples. — Find  the  distances  between  the  following  points  taken 
two  and  two:  (3,  5);  (2,  6);  (—3,  —2);  (—1,  4);  (—2,  — 1); 
C-5,-7);  (-3,0);  (0,-4);  (0,0);  (-5,0). 


THE  BIGHT  LI^'E   IN  A  PLANE. 


19 


SECTION  IV, 

The  Eight  Line  in  a  Plane, 

30,  Bep. — The  Equation  of  a  Locus  is  an  equation  which 
expresses  the  relation  between  the  co-ordinates  of  every  point  in  the 
locus. 


31,  JPvopm    The  Equation  of  a  Eight  Line  passing  through  two  given 


points  is  J  —  y' 


77  (2:  —  x'),  in  which  (x,  y)  is  any  point  in  the 


X  — x 

line,  and  (x',  y')  and  {x'\  j")  are  the  given  points. 

Dem. — Let  M  N  be  any  right  line  referred  to  the  rect- 
angular axes  XX',    YY'.     Let  P  be  any  point  in  the 
line,  and  designate  its  co-ordinates,  AD  and  PD,  by  a; 
andy.     Let  P'  and  P"  be  the  given  points  whose 
co-ordinates  are  x',  t/',  and  x",  y",  respectively. 
Now  drawing    P'E  and  P"  F  parallel  to    AX,       x 
the  triangles  PEP' and  P' FP"  are  similar,  and        m 
give  PE  :  P'F  : :  P'E  :  P"F.  But  PE.  =  y—y\ 
P  F  =y'  —  y",  P'E  =  x  —  x',    and   P" F  = 
as'  —  x" ',  hence,  substituting  these  values,  we  ha-vte 


y—y  '■  y—y 


X  :  X 


or  y  —  y  = 


y 


TT  (a;" —  x).    Q.  E.  D. 


Cor.     1.— Since    P'P'F  =  NGX,    and 


y 


X 


X 


P'F 
P    F 


V  — y 

tan  P'P"F,  wehave^-7 =  the  tangent  of  the  angle  which  the  line 

X   — X 

makes  with  the  axis  of  abscissas. 


If  x' 


X 


n  y'  —  y"  __  y'  —  y" 


.    ,         „  ^        —  GO,  which  being  the  tangent  of 

•JO     iv  vl 

90°,  shows  that  the  line  is  perpendicular  to  the  axis  of  abscissas. 
This  is  as  it  should  be,  since  if  x'  =  x",  the  points  P'  and  P"  are 
equally  distant  from  Y  Y',  and  hence  M  N  is  perpendicular  to  XX'. 


If  y   =    y", 


y'-y"  0 


0,  which  being  the  tangent  of  0°, 


X'  X"  X' X" 

shows  that  the  line  is  parallel  to  (makes  no  angle  with)  the  axis  of 
abscissas.  This  is  as  it  should  be,  since  by  obserying  the  figure,  it 
appears,  that  when  y'  =  y",  M  N  is  parallel  to  XX'. 

32.  Cor.  %—The  Equation  of  a  Might  Line  passing 
through  one  given  point.    If  a;'  =x",  and  y'  =  y",  we  have 


20 


THE  CABTESIAN  METHOD  OF  CO-ORDINATES. 


y  —  2/'  =  0  (^  —  '^0»  ^^^»  by  putting  the  indeterminate  expression, 
^,  =  a,  2/  —  y'  =  a{x  —  x').  This  is  the  equation  of  a  straight  line 
passing  through  a  given  point,  since  the  conditions,  x'  =  x",  y'  =  y"j 
make  P'  and  P"  coincide.  The  a  is  indeterminate,  as  it  should  be, 
since,  through  one  given  point,  an  indefinite  number  of  straight  hues 
can  be  drawn. 

33.  Cor.  3. — 17ie  Common  Equation  of  a  Right  Line, 

If  my  —  y  =  a{x  —  a:'),  we  make  x'  =  0,  and  designate  the  corres- 
ponding value  of  y'  by  h,  so  that  the  given  point  shall  be  the  point  in 
which  the  line  cuts  the  axis  of  ordinates,  we  have,  after  reduction, 
y  ==  ax  -{-  b,  which  is  the  common  equation  of  the  straight  hne.  In  this 
equation  a  is  the  tangent  of  the  angle  which  the  line  makes  with  the 
axis  of  abscissas,  and  b  the  distance  from  the  origin  to  where  the  line 
intersects  the  axis  of  ordinates. 

ScH.  1. — Discussion  of  the  Equation  y  =  ax+b.  If  5  be  -f*  the  line 
cuts  the  axis  of  ordinates  above  the  origin  ;  if  — ,  below  ;  if  0,  at  the  origin. 
In  the  latter  case,  we  have  y  =  ax,  as  the  equation  of  a  right  line  passing 
through  the  origin.  If  a  be  -f,  the  line  makes  an  acute  angle  with  the  axis 
of  abscissas,  [i.  e.,  it  incHnes  to  the  right,  as  the 
lines  in  Fig,  22),  the  tangent  of  an  acute  angle 
being  4--  If  «  be  — ,  the  angle  is  obtuse,  [i.  e. , 
the  Hne  inclines  to  the  left,  as  in  Fig.  23),  since  the 
tangent  of  an  obtuse  angle  is  — .  If  a  =  0  the 
Hne  is  parallel  to  the  axis  of  abscissas,  and  if  a  = 
00 ,  it  is  perpendicular,  as  will  readily  appear. 


X  = 


^H  2. 
1 


-If  we  solve 
b 


in  which  —  is  the 
a 


y  =ax  -\-h  for  x,  we  have 
tangent  of    the 


angle   which  the  Hne  makes  with  the    axis    of 
ordinates,  since,  in  Fig.  21,  the  angle  AHG  = 


NHY  =90°  —    NGX 


In  this 


form,  -  is 
a 


the  distance  on  the  axis  of  abscissas  from  the 

origin  to  where  the  Hne  cuts  it  (AG),  since  the 

base  of  a  right-angled  triangle  is  equal  to  the  perpendicular  divided  by  the 

tangent  of  the  angle  at  the  base. 

34:,  CoR.  A.—TJie  Equation  of  a  Hight  TAne  referred 
to  oblique  axes.    If  the  axes  are  oblique,  we  stiU  have  the  same 


y' 


-,  or  a,  signifies  the  ratio  of  the  sines  of 


forms,  but  in  this  case 

X'  —  X' 

the  angles  which  the  line  makes  with  the  axes,  since  the  sines  of  the 


THE  EIGHT  LINE  IN  A  PLANE. 


21 


angles  of  a  plane  triangle  are  to  each 
other  as  the  sides  opposite.  Thus  in 
Fig.  24,  P'  F  [ory'—y") :  P"F  (or ;r'-^") 
::  AH  :  AG   :  :  sin  AQH  :    sinAHG. 

Putting  /?  for  the  angle  included  by  the      ^^ 
axes,  and  a  for  the  angle  which  the  line 
makes  with  the  axis  of  abscissas,  we  get 


sm  a 


-,  and,  finally,  y  = 


sm  (X. 


y'  —  y"  __ 

x'  —  x"  ~  sm(/i— a)'  "'"^'  —J,  ^  —  g-j^  ^^^ 

tion  of  a  right  line  referred  to  oblique  axes. 
Ex.  1.  Construct  the  equation  y  =  2x  -\-  d. 


Fig.  24 


-a) 


■X-}-  by  as  the  equa- 


SoLUTioNs. — There  are  three  methods  of  solution.  1st.  Bp  any  two  points.  As 
it  is  known  to  be  an  equation  of  a  right  line  from  its  form,  if  any  two  points  be 
determined,  as  in  the  last  section,  the  position  of  the  Une  will  be  known.  For  ex- 
ample, for  a;  =  3,  y  =  9,  and  for  x  =  — 2,  y  =  — 1  ;  whence,  locating  these  points 
and  drawing  a  hne  through  them,  we  have  the  construction. 

2nd.  By  the  intersections  with  the  axes. — -This  is  only  a  modification  of  the  1st 

method,  merely  making  y  =  0,  whence  x  =  — li,  and  making  a;  =  0,  whence 
y  =  3,  constructing  these  intersections,  and  passing  a  line  through  them.  (The 
pupil  should  execute  the  figures.) 

3rd.  By  means  of  the  tangent  of  the  angle  which  the  Une  makes  with  the  axis  of  ab- 
scissas. Since  &  =  3,  we  may  lay  off  A  C  =3  above  the  origin,  and  thus  determine 
C  as  a  point  in  the  line.  Through  C  drawing  C  E  parallel  to  AX  and  constructing 
the  angle  N  C  E  so  that  its  tangent  shall  be  2  (by 
taking  CD  any  convenient  length  and  erecting 
the  perpendicular  FD  =2CD),  the  line  NM 
is  the  one  sought. — Or,  having  located  C,  take  AG 

AC 
=   iAC,  whence  tan    AGC=— r--— -=  2,  will 

AG 

give  the  construction.     Or,  again,  drawing  any  Hne 

making  with  XX'  an  angle  whose  tangent  is  2,  and 

drawing  a  line  parallel  to  it  through  C,  the  latter 

wiU  be  the  hne  sought. 


Y 
C 

/ 

/" 

J 

D               E 

"•  ^ 

A 

X 

Fig.  25. 


ScH. — If  the  tangent  were  — ,  CD  would  be  laid  off  to  the  left  of  O,  or 
the  perpendicular  FD  let  faU  below  D. 

Ex.  2.  Produce  the  equation  of  a  line  passing  through  ( — 3,  5),  and 
(2,  -1), 

Solution. — Here  x'  = — 3,  x"  =2,  y'  =  5  andt/"  =  — 1.     Now,  substituting 


_J/ 


y 


-  (x  —  »'),  and  reducing  to  the  form  y  =ax  -j-h. 


these  values  in  v  —  y' 

X   —  X 

we  have  y  =  —  1.2«  -f-  1.4.      The  pupil  should  construct  this  equation,  and  then 


22  THE  CAETESIAN  METHOD  OF  CO-ORDINATES. 

verify  the  result  by  locating  the  points  ( — 3,  5),  and  (2, — 1),  observing  that,  if  the 
■work  is  correct,  they  will  fall  in  the  line.  Algebraically,  we  verify  the  result  by  sub- 
stituting in  the  equation  y  ■=  —  1.2a;  +  I-^j  successively  for  x  and  y,  ( — 3,  5),  and 
(2,  — 1),  each  of  which  must  satisfy  the  equation,  as  it  expresses  the  relation  between 
the  co-ordinates  of  any  point  in  the  hne.  Substituting,  we  get  5  =3.6  -f-  1.4, 
and  — 1  =  —  2.4  -j-  1.4,  both  of  which  are  correct. 

Ex.  3.  What  angle  does  the  line  which  passes  through  the  points 
(3,  5),  and  ( — 7,  2)  make  with  the  axis  of  abscissas? 

Arts.,  16°  42'  nearly. 

Ex.  4.  Produce  the  equation  of  a  hne  passing  through  the  point 
(2,  — 3),  and  making  an  angle  with  the  axis  of  abscissas  whose  tan- 
gent is  4.  Ans.,  y  =  4:X  —  11. 

Ex.  5.  Produce  the  equation  of  a  line  passing  through  ( — 1,  0),  and 
( — 4,  — 5),  construct  by  the  3rd  method,  and  verify  the  equation  by 
locating  the  points. 

Ex.  6.  Construct  the  triangle  the  equations  of  whose  sides  are  y  == 
|.J7  +  3,  1/  =  —  l-r  4-  4  and  y  ==  ^x  —  1. 

Ex.  7.  What  is  the  equation  of  a  line  which  cuts  the  axis  of  ordi- 
nates  at  3  above,  and  the  axis  of  abscissas  at  5  to  the  left  of  the  origin? 
(Notice  that  this  is  a  case  of  a  line  passing  through  two  points.) 

Ans.,  y  =  S.X  ^  3. 

Ex.  8.  What  hne  ia  y  =  0.x  ?  What  is  x  =  0.y^  How  is  y  = 
0.x  +  4  situated ?     How  y  =  0.x  —  5^ 

Ex.  9.  Find  the  angles  which  the  following  lines  make  with  the 
axis  of  abscissas  :  viz.,  the  hne  passing  through  (3,  5),  and  ( — 1,  — 4); 
through  (5,-2),  and  (5,  3);  through  (—3,  2),  and  (7/2).  How  are 
these  lines  severally  situated  ? 


3S,  I^vop.  Every  Equation  of  the  First  Degree  between  two  variables 
is  an  equation  of  a  right  line. 

Dem. — Every  such  equation  may  be  put  in  the   form  Ay  -^  Bx-\-  C  =  0,  in 

which  A  and  B  are  the  collected  coefficients  of  y  and  x,  and   C  is  the  sum  of  the 

B            0 
absdlute  terms.     By  transposition  and  division  we  have  t/  = —x .  Now 

B  C 

putting  —  —  =  a  and T  ==  ^'  there  results  the  known  form  2/  =  aa;  -{-  6.  q.  e.  d. 

ScH. — If  B  and  A  have  like  signs,  the  line  makes  an  obtuse  angle  with 
the  axis  of  abscissas  ;  and  if  they  have  unlike  signs,  it  makes  an  acute  angle. 
li  B  =z  0  the  Hne  is  parallel  to  the  axis  of  abscissas,  and  if  J.  =  0  it  is 
perpendicular.  If  A  and  C  have  like  signs,  the  line  cuts  the  axis  of  ordi- 
nates  below,  and,  if  unUke,  above  the  origin.     If  (7  =  0  the  hne  passes 


OF  PLANE  ANGLES,   AND  THE  INTERSECTION  OF  LINES. 


23 


through  the  origin.     In  general,  if  an  algebraic  equation  has  no   absolute 
term,  the  locus  passes  through  the  origin.     (Why  ?) 


Ex.  1.  Keduce 


X 


y 


—  4t  =  2x  —  lij 


X 


to  the  form  Ay  -f- 


3  -       —       -^^  ^ 

JBx  4-  C  =  0,  and  describe  the  line   according  to  the  suggestions  in 
the  preceding  scholium. 

Ans. — The  equation  is  y  —  13x  —  21  =  0.  A  =  1,  B  :=  — 13, 
and  0  =  — 21.  As  A  and  B  have  unlike  signs  the  line  makes  an 
acute  angle  with  the  axis  of  abscissas,  the  tangent  of  which  is  13.  It 
cuts  the  axis  of  ordinates  above  the  orisfin  at  a  distance  of  21. 


Ex.  3.  In  like  manner  discuss  3  — 


2  —  y       ^  +  ?/     ^  +  '^y 


—  lOy 


Gx  -j-  y    X  —  V     ,    o  3^  +  ?/ 

— F— ;  — ^—  +  2  =  ?/  H — - — . 


Ex.    4.     Construct  the  figure   the  equations  of   whose    sides    are 
2y  +  2x  =3^  +  3  +  2/;   ^^ —  U  =  2x  —  6  —  y  ;  dy+2x—6 


2y 


X 


+  1 ;  and  x-{-y  =  —  3.     "What  is  the  figure  inclosed  ? 


-♦-♦-«»- 


8JECTI0N  V. 


Of  Plane  Angles,  and, the  Intersection  of  Lines. 

3G»    JPvop,     The  expression  for  the  value  of   an  angle  included 

between  two  lines  is  tan  V  = ~,  in  which  V  is  the  angle  included 

1  4-  aa' 

by  the  lines,  and  a  and  a'  are  the  tangents  of  the  angles  which  the  lines 

make  vjith  the  axis  of  abscissas. 

Dem. — Let  MN  and  M'N',  Fig.  26,  be  two  lines 
whose  equations  are  respectively  y  =:  ax  -^  h  and 
y==  a'x-\-  h'.  Now  C  BX  being  exterior  to  the  triangle 
BCD,  we  have  DCB  =  CBX  — CDB,  or  by 
trigonometry 

tan  CBX  —  tan  CDB 


tanDCB  —  i  ^  ^^n  CBX  X  tan  CDB 

But  DCB  —V,  tanCBX=a',  and  tan  CDB 


==  a.     .-.  tan  V: 


1  +  aa'' 


Q.  E.  D. 


X'    D 


FiQ.  26. 


24  THE  CARTESIAN  METHOD  OF  CO-ORDINATES. 

ScH. — ^In  applying  this  formula  to  any  particular  example,  we  may  obtain 

two  results,  numerically  equal,  but  with  opposite  signs.     Thus,  if  the  two 

lines  are  3/  =  2a7  +  4,  and  y  =  3a;  —  5,  and  we  let  a' =2,  and  a  =  3,  we 

2—3  1 

have   tan  V  =  - — , — ;:,  =  —  --,     But,  if  we  let  «   =  2,  and  a'  =  3,  we  have 
1    +  t)  7 

XT-        3  —  2        1 
tan  V  =  -.        -T  =  ij-     This  ambiguity  is  as  it  should  be,  since  the  two  lines 

form,  in  general,  two  equal  acute,  and  two  equal  obtuse  angles  with  each 
other  ;  and  as  these  angles  are  supplements  of  each  other,  they  have  tan- 
gents numerically  equal  but  with  opposite  sigyis. 


37 »  J*VOb.     To  find  the  equation  of  a  line  which  makes  any  required 

angle  loith  a  given  line. 

Solution. — Let  y  =  ax  -{-hloe  the  equation  of  the  given  line,  y  =  a'x  -j-  h'  he 
that  of  the  required  hne,  and  m  the  tangent  of  the  required  angle.  As  the  relative 
directions  of  the  hnes  depend  solely  upon  a,  a',  and  m,  the  problem  consists  in 

finding  the  unkLio\\Ti  a',  in  terms  of  the  given  tangents  a  and  m.     But  m  =  — -; 

I  -\-  aa 

by  the  preceding  proposition;   whence  a'  = :  andv== — ~ tc  +  feis 

the  equation  of  the  required  Hne. 

ScH. — In  this  form  b'  is  undetermined,  as  it  should  be,  since  there  maybe 
an  indefinite  number  of  lines  which  will  satisfy  the  condition,  all  having  the 
same  inclination  to  the  axis  of  abscissas,  but  cutting  the  axis  of  ordinates 
at  different  points. 

CoK.  1. — If  the  required  line  is  to  pass  through  a  given  pioint  (x',  y'),  ice 

,  a  4-  m     , 

have  J  —  y    =   -; (x  —  x'). 

•^  -^  1  —  am  ^  ^ 

38,  CoE.  2. — If  the  required  line  is  to  be  parallel  to  the  given  line, 
m  =  0,  and  we  have  ai,'=  a.  The  equations  then  become  y  =  ax-\-  b', 
and  y  —  y'  =  a{x  —  x')^  both  of  which  lines  are  parallel  to 
y  =  ax  -]-  h. 

30,  CoR.  3. — If  the  lilies  are  to  be  perpendicular  to  each  other,  m  =  00. 

a  -\~  m  in  1 

.'.  a'  =  - :=, ^  = ,  or  1  +  act'  =  0,  which  is  called   the 

1  —  am       — am  a 

equation  of  the  condition  of  peiyendicularity.      The  equations   of  lines 
j)erpendioular  io  y  =  ax  -{-  b  will  therefore  be  y  =  ■ x  +  I'',  and 

y  —  y'  = (^  — ^'),  the  latter  passing  through  {x',  y'). 

*  The  principle  upon  which  this  reduction  is  effected  is,  that  the  finite  terms  a  and  1  added  to  the 
infinites  m  and  — am  must  be  dropped.  The  axiom  is,  Suites  added  to  infinites  do  not  (apprecia- 
bly) affect  the  ratio  of  the  infinites.  The  word  appreciably  is  throwu  in  to  aid  the  student's 
apprehenaion.    It  is  not  required,  nor  is  it  strictly  correct. 


OF  PLANE   ANGLES,   AND   THE  INTERSECTION   OF  LINES.  25 

ScH.  2. — Two  lines  are  parallel  to  each  other  when  the  two  equations  being 
reduced  to  the  form  y=a.c(7  +  Z>,  the  coefficients  of  a;  are  the  same  in  both; 
and  they  are  perpendicular  when  these  coefficients  are  reciprocals  of  each 
other  with  opposite  signs. 

Ex.  1.  Find  the  angle  included  between  y  = — ^  +  2,  and  y  =  dx 6. 

3  +  1 
Besult,    Tan  V=  fTTQ  '=~^-     •*•    ^^®  ^^S^^  is  ll6°34^ 

Ex.  2.  What  are  the  angles  of  the  triangle  the  equations  of  whose 
sides  are  2y  —  5  =  y  —  x;  y  -^  4:X=8,  and  y=zix? 

Ans.  I    '^^®  tangents  of  the  angles  are  .6,  —21,  and  1.5. 
*  1    The  angles  are  nearly  30°58',  92°44',  and  56°19'. 

Ex.  3.  Write  the  equations  of  three  lines,  each  parallel  to  y  =  2x 

11,  and  construct  the  lines  therefrom. 

Ex.  4.  Write  the  equation  of  a  line  parallel  to  y  —  i,r  =  5  and 
passing  through  (—6,  4).  Keduce  the  equation  of  the  parallel  to  the 
form  y  =  ax  -{-  b,  and  then  construct  both  lines  from  their  equations. 
Verify  the  result  by  constructing  the  given  point  ;  also  by  observino- 
that  the  ]  coefficients  of  x  in  both  equations  are  equal,  and  that  the 
co-ordinates  of  the  given  point  satisfy  the  equation  of  the  parallel. 

Ex.  5.  Write  the  equation  of  a  line  passing  through  ( —  ^,  4),  and 
parallel  to  ^x  —  ^y  =  2.     Verify  as  in  Ex.  4. 

Besult  The  equation  is  ?/  =  f^  -f  1^. 

Ex.  G.  Write  the  equations  of  three  lines  each  perpendicular  to 
^y  —  2x  =  1,  reduce  them  to  the  form  y  =  ax  -{-  b,  and  verify  the 
results  by  construction. 

Ex.  7.  Write  the  equation    of  a  line  perpendicular  to  2y 4=  =  x 

and  passing  through  (1,  — 3).     Verify  as  before. 


Ex.  8.  Write  the  equations  of  lines    perpendicular  to 


x^y 


.^J7  — 2,  and  severally  passing  through    (—2,3);    (0,-5);    (0,0); 
and  (—3,  0). 

Ex.  9.  What  is  the  angle  included  between  y  =  0.07,  and  y  =  3x 
—  5  ?     Between  x  =  O.y,  and  y  =  2x  -{■  1? 

Ex.  10.   What  is  the  equation  of  a  line  passing  through  ( 6,  0), 

and  perpendicular  to  y  =  0.x  +  5  ?  Ans.,  x  =  O.y 6. 


26  THE   CARTESIAN   METHOD   OF   CO-OEDINAIES. 

Ex.  11.  "What  is  the  equation  of  a  line  passing  through  ( — 1,  3),  and 
making  an  angle  of  45°  with  y  ^=.^x  —  5  ?  ^ns.,  y  =  — 3a:. 

Ex.  12.  Produce  the  equation  of  a  hne  passing  through  ( — 4,  — 5) 
and  making  an  angle  of  71°  34'  with  y  ==  — 2a:  +  7  ? 

Besult,  y^=\x  —  4f,  calling  tan  71°34',  =  3. 


4:0 •  Pvohm     To  find  the  point  or  points  of  intersection  of  two  lines. 

Solution. — For  a  common  point  tlie  values  of  x  and  y  are  the  same  in  both 
equations,  and  only  for  such  a  point  Therefore,  making  the  equations  simultaneous 
restricts  the  values  of  x  and  y  to  the  required  point  or  points.  Consequently,  we 
have  only  to  solve  any  two  given  equations  for  the  values  of  x  and  y  in  order  to 
find  the  point  or  points  in  which  the  loci  intersect. 

ScH.  1. — The  general  formulcR  for  the  value  of    the  co-ordinates  of  the 

point  of  intersection  of  two  straight  Unes  whose  equations  are  y  =^  ax  -{-h^ 

h'  —  h                    ah'  —  ah 
and  V  =  ax  +  &',  are  x  = :,  and  y  = ■, — .      Upon  these  values 

^  a  —  a  ^  a  — a  ^ 

we  may  observe  that  for  a  =  a',  and  b  and  b'  unequal,  the  values  of  x  and 

y  become  oo.      This  indicates  that  the  hnes  do  not  intersect,  and  hence  that 

they  are  parallel.     Therefore  a  =  a'  is  the  condition  of  parallelism  of  two 

straight  lines.     This  may  also  be  seen  directly  from  the  meaning  of  a  and  a. 

As  these  quantities  are  the  tangents  of  the  angles  which  the   lines  make 

with  the  axis  of  abscissas,  it  follows  that  when  they  are  equal  the  lines  make 

equal  angles  with  this  axis,  and  are,  therefore,  parallel.     2nd.  Jlh  =  h',  and 

a  and  a  are  unequal,  we  have  x  =  0,  and  y  =  h  =:^h' .     This  is  also  evident 

from  the  meaning  of  h  and  h' .     Both  lines  cut  the  axis  of  ordinates  at  the 

same  point.     3rd.    If  a  =z  a  and  h  =  h' ,  x  =%,  and  y  =  %,  and  the  Hnes 

coincide.    4th.  If  a  =  0,  and  6  =  0,  the  first  equation  becomes  ?/  =  O.a;  -f  0> 

h' 

or  the  equation  of  the  axis  of  abscissas,  and  x= -,  the   pomt   of    inter- 

a 

section  oiy  =  a'x  -f  h'  with  this  axis,  as  it  ought.     Sch.  2,  Art.  33. 

SCH.  3.— Any  two  equations  between  two  variables  being  given,  if  the  lines 
they  represent  are  constructed,  and  the  co-ordinates  of  the  points  of  intersection 
measured,  we  have  a  graphic  solution  of  the  equations. 

Ex.  1.  Where  is  the  point  of  intersection  of  the  lines  ^x  — \y=^\ 
and  2/  =  —  2a:  4-  4 ?  Ans.,  (2,  0). 

Ex.  2.  What  are  the  co-ordinates  of  the  vertices  of  the  triangle  the 

equations  of  whose  sides  are  y  —  2a:  +  3  =  0,    — ^— ^  +  4:  ==  ^y,  and 

a:  —  iy  =  2  ? 

Ex.  3.  Where  does  a  perpendicular  from  ( — 3,  8),  to  the  line  3/  = 
\x  — 5,  intersect  the  latter?  An^^  At  (l-J,  — 4|-). 


OF  PLANE  ANGLES  AND   THE  INTEBSECTION  OF  LINES. 


27 


Ex.  4.    "VVlaere  does    a   perpendicular    from  the    origin    intersect 
2ar  — 3?/  =  4? 

Ex.  5.  Given  y  =  ^x  —  3,  y  =  — 4ar  —  8,  and  y  =  — ^x  +  10, 
as  the  equations  of  the  sides  of  a  triangle,  required  to  find  where  a 
perpendicular  from  the  angle  included  between  the  first  two  sides, 
intersects  the  third  side. 

Besult,  At  the  point  5 .5,  6. 34,  nearly. 

Ex.   6.    Eind  the   intersections    of    the   loci  y 

whose  equations  are  7{y  —  x)  =  5  —  2x,  and 

2/2   +   ^2     _|_     9   =:::=  Ig  Qy^     q^j^^    COUStrUCt 

the  figure. 

Results,  At  the  points  (.374  +,  .981  +),        P^J- 
and  (—3.888  +,  —2.063  +).      The  figure     M 
is  that  given  in  the  margin. 

Ex.  7.  Eind  the  intersections  of  y^  = 
IQx,  and  x^  -j-  t/2  =  144,  and  construct 
the  loci,  thus  verifying  the  solution. 


Fig.  27. 


Ex.  8.  Eind  the  intersections  of  25?/2+  IBar^  =  1600,  and  IQy^ —  Oo;' 
+  576  =  0.       Results,  At  (9.12,  3.3),  and  (—9.12,  —3.3),  nearly. 

Ex.  9.  Eind  the  intersections  of  x"^  —  Sx  -\-  y"^  -\-  Qy  =^  0,  and  y  = 
\x  -\-  1.     Also  of  the  first  with  3?/  =  4ar.     Also  with  2/  ^=  3  —  x. 
Results. — 1st.  Imaginary  results.      No  intersections.     2nd.  A  com- 
mon point  at  the  origin.     3rd.  Two  points  of  intersection. 

ScH,  3. — The  construction  of  loci  represented  by  equations  affords  beau- 
tiful illustrations  of  principles  in  the  theory  of  equations,  concerning  the 
number  and  character  of  the  roots  of  an  equation. 

Ex.  10.    Eind    the    intersections  of    25i/2  + 
4a;2  =  100,  by  the  following  :  1st,  y^  + 
a;'  =  9  ;   2d,   y^  +  2y  +  a;^  =  8  ;  3d, 
2/2  +  41/+ ^2  =  5;  4th,  2/2  +  101/ +07'=    y^^^ 
— 16  ;  and  5th,  i/^  +  Vly  +  ^2  =  — 27. 

Results.— Ui,  At  (2.44, 1.7);  (—2.44, 1.7); 
(2.44,  —1.7);  (—2.44,  —1.7)  which  affords 
an  example  of  4  real  roots.  2nd,  At  (0.2)  ; 
and  (=fc  2.9,  — f  f ),  which  affords  an  example 
of  what  seems  to  be  but  three  roots  when  there 
should  he  four.  This  is  explained  by  the 
two  values  of  xior  y  =  2,  becoming  -f  0  and 


B  X 


28 


THE   CARTESIAN  METHOD   OF  CO-ORDINATES. 


— 0,  or  practically,  though  not  theoretically,  one.  3rd,  Gives  two  real 
and  two  imaginary  points,  illustrating  that  imaginary  roots  enter  in 
pairs.  4th,  Gives  two  equal  real  roots,  both  0,  and  two  imaginary, 
showing  a  point  of  contact.  5th,  Four  imaginary  roots,  showing  no 
common  point,  the  additional  imaginary  roots  again  entering  in  a 
pair. 

Ex.  11.  Find  the  intersections  of  1y^  —  Lxy  +  '^x"  —  3?/  —  2x  —  8  = 
0,  by  4?/2+4^2 — 11  =  0.  Also  by  y^-\-2y -^-x'^  —  6a? +  6  =  0.  Also 
by  2/2+6?/+^2 — 4^+9=:  0. 

Results. — By  the  first  in  4  points.  By 
the  second  in  2  points.  By  the  third  not 
at  all.  The  figure  is  seen  in  Fig.  29,  in 
which  a  a  a  is  the  1st  locus,  and  111, 
2  2  2,  and  3  3  3,  the  others,  in  order. 


4:1,  JPvob,     To  find  the  perpendicular 
distance  from  a  given  point  to  a  given  line. 

Method  of  So"littion. — First,  find  the  equation 
of  a  line  passing  through  the  given  point,  and 
perpendicular  to  the  given  Hne  {32 ,  39).  Second,  Fig  29.. 

find  the  point  in  which  this  perpendicular  inter- 
sects the  given  line  {40).    The  problem  then  consists  in  finding  the  distance  between 
two  points  {29. ) 

Cor. To  find  the  distance  between  two  parallels,  wi'ite  the  equa- 
tion of  a  line  perpendicular  to  the  parallels  {39),  and  find  its  intersec- 
tions with  the  parallels.     The  problem  is  then  the  same  as  {29). 

Ex.  1.  Find  the  distances  of  the  following  points  from  each  of  the 
lines  2/  =  2j:  —  3,  and  ^x  —  y  =  — 1,  viz.,  3,  2  ;  — i,  — 1;  0,  — 6  ; 
0,  0. 

Solution. — To  find  the  distance  from  — 4,  — 1,  to  ?/  =  2a;  — 3,  we  have  for  the 
equation  of  a  line  passing  through    this   point  and  perpendicular  to  this  line 

y  _j_  1  = 4 (a;  -}-  4),  or  2/  =  — l^  —  3.      The  intersection  is  at  0,  — 3.      The 

distance  between  —4,  —1  and  0,  —3  is  D  =  >/l6  -}-  4,  or  ^20. 

Ex.  2.  Find  the  sides,  the  angles,  and  the  perpendicular  distances 
from  the  angles  to  the  opposite  sides  in  the  triangle  the  equations  of 
whose  sides  are  36i/  —  4^  =  45,  3?/  +  3  =  —x,  and  j/  =  fa;  —  3. 

f    The  sides  are  12.79,  7.44,  and  6.79. 
' Besults.—  ■]    The  angles  are  24°46',  127°53',  and  27°21'. 
(.  The  perpendiculars  are  5.88,  3.12,  and  5.36. 


OF  THE  CONIC   SECTIONS. 


29 


Ex.  3.  The  yertices  of  a  triangle  are  at  2,  8  ;  — 6,  1  ;  and  0,  — 4  ; 
required  the  equations  of  the  sides,  of  the  hnes  drawn  from  the  vertices 
to  the  middle  of  the  opposite  sides,  and  of  the  lines  drawn  bisecting 
the  angles  and  terminating  in  the  opposite  sides. 


^  ^  ^- 


SECTION  VI 

Of  the  Oonic  Sections. 

42.  Boscovich^s  Definition  of  a  Conic  Section. — 

A  Conic  Section  is  a  curve,  the  distance  of  any  point  in  which  from  a 
given  point,  is  to  its  distance  from  a  given  straight  line,  in  a  given 
ratio.  If  the  distance  to  the  point  is  equal  to  the  distance  to  the  line, 
the  locus  is  a  JParahola  ;  if  less,  an  JEllipse  ;  if  greater,  an 
Hyperbola.  If  the  distance  to  the  line  is  infinite,  the  locus  is  a 
Circle  ;  but  if  the  distance  to  the  'point  is  infinite,  the  locus  is  a 
Straight  Line. 

43.  I*roh.     To  construct  a   Conic  Section  from  Boscovich's   de- 
finition. 

Solution. — Let  F,  Fig.  30,  be  the  given  point, 
A  B  the  given  line,  and  m  :  n  the  given  ratio. 
Through  F  draw  C  K  perpendicular,  and  G  H 
parallel  to  AB.  Take  FG  (=  FH)  :  FC  :  : 
m  :  n,  and  draw  CG  and  CH,  i^roducing  them 
indefinitel3\  Drav/  a  series  of  parallels  to  G  H , 
meeting  the  lines  C  M  and  C  N .  Now  with  the 
half  of  any  one  of  these  Hnes,  as  LT",  for  a  radius, 
and  the  given  point,  F,  as  a  centre,  describe  an 
arc  cutting  the  parallel  taken,  as  at  P.  Then  is 
P  a  point  in  the  curve.  To  prove  that  P  is  a 
point  in  the  curve,  join  P  and  F,  and  draw  PR 
parallel  to  C  K.  By  similar  triangles  we  then  have  Fig.  30. 

LT(=  PF)  :  XC  (  =  PR)  :  :  G  F  :  FC   (by  construction)  :  :  m  :  w. 

P  R  :  :  m  :  ?i.  In  like  manner  any  required  number  of  points  in  the  curve  may  be 
determined,  so  that  by  connecting  them  the  curve  will  be  completely  drawn.  In 
this  figure,  as  FG  <  FC  the  curve  is  an  ellipse.     Had  FG  been  taken  equal  to 

FC,  the  curve  would  have  been  a  Parabola.     And  if  FG  had  been  greater  than 

FC,  the  curve  would  have  been  an  Hyperbola. 

44.  Defs. — The  fixed  line,  AB,  is  the  Directrioc.  The  fixed 
point,  F,  is  the  Focus.  CM  and  CN  are  the  Focol  Tan- 
gents. 


PF 


30 


THE  CAETESIAN  METHOD  OF  CO-OEDINATES. 


The  portion  of  the  perpendicular  to  the  directrix  through  the  focus, 
CK,  intercepted  by  the  curve  is  the  Transverse  or  Jfajor  A-Xis^ 
as  I K.     The  centre  of  the  transverse  axis,  O,  is  the  centre  of  the  curve. 

The  perpendicular  to  the  tranverse  axis  passing  through  the  centre, 
and  limited  by  the  curve  (in  the  ellipse),  as  DE,  is  the  Conjugate, 
or  3£inor  A.xis.  The  double  ordinate  passing  through  the  focus, 
GH,is  the  Latus  MectuTTi,  I*rincipal  I^arameter,  or  the 
parameter  to  the  transverse  axis.  The  extremities  of  the  transverse 
axis,  I  and  K,  are  the  Vertices.  The  distances  from  the  focus  to  the 
vertices  are  the  Focal  Distances.  The  JEccentricity,  is  the 
distance  from  the  focus  to  the  centre,  divided  by  the  semi-transverse 


axis. 


FO 
lO 


Ex.  1.  Construct  a  parabola  whose  parameter  is  12. 


CoNSTEUCTioN.  —Let  A  B  be  the  directrix.  Draw 
C  K  perpendicular  to  it,  take  F  at  a  distance  6  from 
the  directrix,  and  through  F  draw  G  H  parallel  to 
A  B.  Take  G  F  =  H  F  =  G,  and  draw  C  M,  and 
C  N  through  G  and  H .  (The  construction  is  then 
completed  as  in  the  problem  above. )  To  show  that 
any  point  thus  found,  as  P,  is  a  point  in  a  parabola 
whose  parameter  is  12,  observe  that  LT  (=PF): 
CT  (=  PR)  ::GF  :CF.  ButGF  =  CF  =  6, 
by  construction.  .-.  PF  =  PR.  Also  GH, 
the  parameter, *=  2G  F  =  12. 

QuEEiES. — Can  the  parabola  ever  return  into  itseK 
so  as  to  inclose  a  space  ?  "Why  ?  Can  j)ortions  of  this 
curve  lie  on  both  sides  of  the  directrix  ?     Why  ? 


r        K 


Fig.  31. 


4:S.  CoE.  1. — If-^  =^the  focal  ordinate,!,  e.,  one  half  the  Latus  Rectum, 
the  vertex  is  at  ^-pfrom  the  directrix,  and  the  distance,  if  =  ^i^,  or  ^ 
the  parameter,  GH. 

Ex.  2.  Construct  an  ellipse  in  which  the  fixed  ratio  shall  be  ^,  and 
the  distance  from  the  focus  to  the  directrix  6.  Also  with  the  ratio  |- 
and  the  distance  from  the  focus  to  the  directrix  5. 


QuEBiES, — With  the  same  focus  and  directrix  how  does  varying  the  ratio  affect 
the  form  of  the  ciirve  ?  With  the  same  ratio  and  directrix  how  does  varying  the 
position  of  the  focus  affect  the  form  of  the  curve  ?  How  does  it  appear  from  the 
first  query  that  when  the  ratio  is  0,  the  locus  is  a  point  ? 

Ex.  3.  Construct  an  eUipse  whose  latus  rectum  shall  be  6,  and  the 
fixed  ratio  f . 


OF  THE  CONIC  SECTIONS.  31 

Suo.— In  Fig.  30,  if  Q  F  =  3  and  the  characteristic  ratio  of  the  curve  is  ^,  what 
is  OF? 

46,  CoK.  2 — From  Fig.  30,  by  principles  of  construction  it  appears  that 
the  tangents  at  the  vertices,  viz.,  IS,  and  KM,  are  equal,  respectively,  to 
the  focal  distances  I  F,  and  F  K-  It  also  appears  that  the  distance  from 
the  focus  to  the  extremity  of  the  conjugate  axis  in  an  ellipse,  F  D,  equals 
the  semi-transverse  axis  ;  for  F  D  =  QO  =  -^'l  I S  +  M  K)  =  ^1  K. 

Ex.  4.  Construct  an  ellipse  whose  transverse  axis  shall  be  12,  and 
conjugate  10  ;  i.  e.,  having  given  the  axes,  to  construct  the  ellipse. 

Ex.  5.  Construct  an  ellipse  whose  transverse  axis  shall  be  10,  and 
distance  between  the  foci  8. 

Ex.  6.  Having  given  the  curve  and  the  transverse  axis,  to  find  the 
foci  and  directrix  of  an  ellipse. 

Solution. — Let  ACBC,  Fig  32,  be  the  curve,  and 
A  B  its  transverse  axis.  Bisect  the  transverse  axis  with 
a  perpendicular,  and  the  portion  of  this  perpendicular 
intercepted  by  the  curve  will  be  the  conjugate  axis. 
From  either  vertex  of  the  conjugate  axis  as  a  centre,  with 
a  radius  equal  to  the  semi-transverse  axis,  describe  arcs 
cutting  the  transverse  axis  ;  these  points  will  be  the  foci  Fig.  32. 

{46).    As  there  are  two  intersections,  there  are  two  foci. 

At  each  extremity  of  the  transverse  axis  erect  perpendiculars  and  make  them 
severally  equal  to  the  adjacent  focal  distances,  thus  obtaining  two  points  in  the 
focal  tangent  (46).  Draw  the  focal  tangent,  and  where  it  intersects  the  transverse 
axis  produced,  erect  a  perpendicular  to  this  axis,  and  this  perpendicular  wiU  be  the 
directrix. 

Queries.— How  does  it  appear  from  the  definition  of  the  ellipse,  that  the  curv^e 
can  not  lie  on  both  sides  of  the  directrix  ?  How  does  it  appear  that  the  curve  cuts 
the  axis  beyond  the  focus  ? 

Ex.  7.  Letting  A  represent  the  semi-transverse  axis,  B  the  semi- 
conjugate,  2c  the  distance  between  the  foci,  and  e  the  eccentricity^ 
show  that 

c   v/^2 i?2 

^—A~        I        ' 

B^ 
and  hence  that  1  —  ^^  ^=  "7~ '   How  does  it  appear  from  this  that  in 

the  case  of  the  ellipse  e  <^12 

Ex.  8.  Construct  an  ellipse  whose  transverse  axis  is  12,  and  eccen- 
tricity |-. 

SuG.     First  find  the  value  of  B,  which  is  d^-,  nearly. 


32 


THE  CABTESIAN  METHOD  OF  CO-ORDINATES. 


Ex.  9.     Construct  an  hyperbola. 


Fig.  33. 


SoiiUTioN. — Let  AB  be  the  directrix, 
F  the  focus,  and  m  :  n  the  ratio,  in 
which  m  >>  n.  Through  F  draw  FK 
perpendicular  to  the  directrix  and  GH 
parallel,  producing  both  indefinitely. 
Take  FG  (=  FH)  :  CF  :  :  rn  :  n. 
Through  C  and  G  draw  MM',  and 
through  H  and  C,  NN',  the  focal  tan- 
gents. (The  process  is  exactly  analogous 
throughout,  to  that  pursued  in  construct- 
ing the  ellipse,  and  hence  need  not  be 
detailed.  The  student  can  supply  it.  It 
should  be  noticed,  however,  that  the 
distance  from  the  focus  to  any  point  in 
the  curve,  being  greater  than  the  distance 
from  the  same  point  to  the  directrix,  there 


may  be  (are)  points  in  the  cui've  on  the  opposite  side  of  the  directrix  from  the  focus. 
These  points  are  determined  in  the  same  manner  as  the  others.  Thus  the  point 
P'^"  is  found  by  taking  T'N'  as  a  radius,  and  from  F  as  a  centre  drawing  an  arc 
cutting  T'N'  in  P^'"^.     In  like  manner  other  points  in  this  branch  are  located.) 

The  demonstration  is  as  foUows  :  To  prove  that  any  point,  as  P ,  is  in  the  curve, 
we  have  to  prove  that  PF  :  PR  : :  wi  :  7i  ;  {.  e.,  the  distance  from  any  point  in 
the  curve  to  the  focus,  is  to  the  distance  of  the  same  point  from  the  directrix,  in  a 
constant  ratio  (m  :  n),  which  ratio  is  greater  than  1,  in  the  hyperbola.  To  prove 
that  the  construction  gives  this  proportion,  join  P  and  F,  and  draw  PR.  parallel 
to  TC.  Now  since  PF  =  LT,  and  PR  =:  TC,  and  by  reason  of  similar  trian- 
gles, we  have  PF  :  PR  : :  LT  :  TC  *• :  GF  '•  FC  : :  w  :  n.  In  a  similar  manner 
any  point  on  the  other  side  of  the  directrix,  found  by  the  method  described,  as 
P^ii,  is  shown  to  be  in  the  curve.  Thus  pviip  ^^^  N'T'  by  construction,  P"^"R'  = 
T'C,  and  the  triangles  CFG  and  CT'N'  are  similar  ;  hence  pv^F  :  P^"R'  : : 
T'N'  :  T'C  : :  G  F  :  FC  :  m   :  n.     Q.  E.  D. 

4:7.  Def's. — Tlie  Aocis  of  the  Hyperbola  is  an  infinite  line 
drawn  through  the  focus  and  perpendicular  to  the  directrix,  as  TT' 
Fig.  33. 

The  Transverse  Aocis  of  the  Hyperbola  is  that  por- 
tion of  the  axis  of  the  curve  included  between  the  vertices,  as  K I , 
Fig.  33. 

TJie  Focal  Distances  are  the  distances  from  the  focus  to  the 
yertices,  as  Fl,  and  F  K,  Fig.  33. 

TJie  Conjugate  Aocis  of  the  Hyperbola  is  a  perpendic- 
ular to  the  transverse  axis  at  its  centre,  and  is  limited  by  an  arc 
drawn  from  the  vertex  as  a  center,  with  a  radius  equal  to  the  distance 
from  the  focus  to  the  centre.     Thus,  in  Fig.  33,  D  E  represents  the 


OF  THE  CONIC  SECTIONS. 


33 


conjugate  axis,  the  extremities  D  and  E  being  determined  by  making 
the  distances  Dl  and  El  each  equal  to  OF-  This  definition  is  a 
convention  adopted  for  the  purpose  of  rendering  more  close  the  ana- 
logy between  this  curve  and  the  ellipse. 

A  Conjugate  Hyperbola  is  an  hyperbola  having  the  conju- 
gate axis  of  a  given  hyperbola  for  its  transverse  axis,  and  the  trans- 
verse axis  of  the  given  curve  for  its  conjugate ;  see  Ex.  10,  F%g.  34. 
Either  of  two  hyperbolas  thus  related  is  conjugate  to  the  other.  They 
are  sometimes  distinguished  as  the  X  hyperbola  and  the  Y  hyper- 
bola, each  taking  the  name  of  the  co-ordinate  axis  upon  which  its 
transverse  axis  Hes. 

An  Equilateral  Hyperbola  is  one  which  has  its  conjugate 
-axis  equal  to  its  transverse. 

Ex.  10.  To  construct  a  pair  of  conjugate  hyperbolas  whose  axes  are 
8  and  6. 

SuGS. — Draw  two  indefinite 
straight  lines  at  right  angles  to 
each  other,  and  take  Ol  == 
OK  =4,  andOD=OE=3. 
Having  constructed  the  branches 
on  the  axis  Kl,  Fig-  34,  as  in 
Ex.  9,  take  O F'  =  OF  (which 
=  ID),  and  F' is  the  focus  of 
the  conjugate  or  Y  hyperbola. 
Taking  DS'  =  D  F'  and  E  L'  = 
EF',  and  through  S'  and  L' 
drawing  a  right  line,  it  is  one 
of  the  focal  tangents.  Having 
found  the  focal  tangents  the 
construction  proceeds  as  before. 


Fig.  34. 


Ex  11.  Construct  an  hyperbola  whose  transverse  axis  is  G,  and 
less  focal  distance  2.  Eind  also  the  conjugate  axis,  focus,  and  direc- 
trix of  the  conjugate  hyperbola. 

Ex.  12.     Letting  e  represent  the  eccentricity  of  an  hyperbola,  c  the 
distance  from  the  centre  to  the  focus,  A  the  semi-transverse  axis,  and ' 
B  the  semi-conjugate,  show  that 


1^' 


Why  is  e 


e=z-  =  id!±J?!)I    and  hence  that  1  —  e»  =  — 

A  A 

greater  than  1  in  the  hyperbola? 

Ex.  13.     "What  is  the  eccentricity  of  an  hyperbola  whose  axes  aro 


i 


34 


THE  CARTESIAN  METHOD  OP  C0-0EDINATE8. 


10  and  6  ?    Wliat  is  the  eccentricity  of  an  hyperbola  whose  transverse 
axis  is  12,  and  less  focal  distance  3? 

Ex.  14.  The  eccentricity  being  1^  and  the  conjugate  axis  4,  what 
is  the  transverse  axis  ?  What  the  focal  distances  ?  What  the  charac- 
teristic ratio  {4:2)  ?  Transverse  Axis,  3.577  +. 


4=S»     ^TOp,     Boscovich's  ratio  and  the  eccentricity  are  equal. 

Dem. — 1st.     Let  AB,  Fig.  35,  be  the  directrix 

of  an  ellipse,   F  the  focus,  CL  the  focal  tangent, 

and   O    the   centre.     Draw  GK  parallel  to  CL- 

Then  since  LO  =  GO,  and  GF  =  IG  =  LK 

"\46),  LO  -  LK  -  KO  =  GO  -  GF  =  FO. 

KO 

Therefore,  — — -  =  the  eccentricity  {44).     By  defi- 

GO 

nition  =  =  Boscovich's  ratio.     Now,  by 

CG        CG  '    ^ 


Pig.  35 


Birailar  triangles, 


IG 
CG 


KO 
GO 


Q.  E.  D. 

2nd.  In  the  Hyperbola  the  demonstration  is 
essentially  the  same.  Thus,  in  Fig.  36,  LO  = 
EM   -    IG        FE  —  FG 


2 
LO  +  LK  =   KO 
KO 


=  GO. 


Hence 


and 


GO 
GF  IG 

CG 


GO  +   FG   ==  FO, 
=   the    eccentricity.      By    definition 


=   the   characteristic   ratio    (Bos- 
Now,     hy    similar    triangles, 

Q.  E.   D. 


Fig.   36. 


CG 

covich's     ratio) 
IG  __  KO 
CG        go' 

3rd.  In  the  Parabola  we  may  call  the  eccen- 
tricity 1,  from  analogy ;  or,  better,  we  may  conceive  the  parabola  to  be  an  ellipse 
with  the  centre,  O,  removed  to  an  infinite  distance  from  the  vertex,  G,  Fvj.  35, 

whence  the  fraction  -— —  =1.*     Q.  e.  d. 
QO 

49,  CoR. — The  student  should  fix  in  memory  the  following  relations, 
as  they  are  fundamental,  and  of  frequent  use  in  the  reduction  offormulw. 

Letting  A  =  the  semi-transverse  axis,  B  =  the  semi-conjugate  axis, 
e  =  the  eccentricity,  and  p  ==  the  semi-latus  rectum,  we  have  the  fol- 
lowing 

*  If  the  student  has  difficulty  in  understanding  this  statement,  let  him  consider  that,  O  being 
roraoved  to  infinity,  the  finite  distance,  GF,  by  which  GO  appears  to  be  greater  than  FO,  is  of  no 
appreciable  vahie  as  compared  vnth.  the  terms  of  the  ratio,  which  are  both  infinite. 


OF  THE  CONIC   SECTIONS. 


35 


POfDAMENTAL  RELATIOIVS. 

From  the  definition  {42)  and  {48),  we  see  that,  The  distance  from 
any  point  in  the  curve  to  the  focus  -^  e  =  the  distance  from  the  same 
point  to  the  directrix.  Also,  The  distance  of  any  point  in  the  curve 
from  the  directrix  x  e  :=  the  distance  of  the  same  point  from  the  focus. 


Distances. 

Ellipse. 

Hyperbola. 

Parabola 

1. 

(  Focus  to  extremity  of  Conj.  -axis  . .   = 
(  Veetex         "                  "         = 

A 

00 

CO 

Ae 

2. 

"Por.TTS  to   riKNT-RF.  .                                                            

Ae 

Ae 

cc 

3. 

Focal  Distances — 

A{1  =F  e) 

A{e  =F  1) 

hp 

4. 

Vertices  to  Dieecteix.  ...         .   — 

^(1  =F  e) 
e 

A(e  ^  1) 
e 

hp 

5. 

Focus  to  Dieecteix — 

A{1  —  e-^) 
e 

A{e'^~l) 
e 

p 

6. 

r.-p.TxTTEE  t*^  DtEK^'TKTX                                            

A 

e 

A 

e 

00 

7. 

1— e^                                           _ 

A^ 

A-^ 

0 

p 

8. 

Semi-Latus  Kectum,  p = 

A 

A 

Dem. — For  the  ellipse  see   Fig.  30,  for   the   hyperbola, 
Mg.  33,  and  for  the  parabola.  Fig.  37. 

(1.)  For  Fllipse  see  {4:6). — For  Hyperbola,   by  definition 

FO 

of  eccentricity  — - —  =  e.      .  • .    F  O  =  ^e  =  the    distance 

from  the  vertex  to  the  extremity  of  the  conjugate  axis,  by 
the  definition  of  the  latter  (47)-  For  Parabola,  consider 
the  curve  as  an  ellipse  with  its  centre  removed  to  infinity. 


FO 

(2.)  For  Mlipse,  — —  =e  by  definition. 


FO  =^e. 


Fig.  37. 


For  Hyperbola  and  Parabola,  see  above. 

(3.)   For  Ellipse,    \F  =  \0  —  FO  =  A  —  Ae  =  A{1  —  e).       FK  =  OK 

-|-FO=^4-^e  =  ^(1  -|-  e).     For  Hyperbola,    IF=  FO  —  \0  =^  Ae  —  A  = 

^(e  —  1).      F K  =  FO  -f  O  K  =  ^e  +  -4  =  ^(e  -f  1).     For  Parabola  see  {4:5). 


(4.)  For  Ellipse,  since  I  is  a  point  in  the  curve    IC= = 

6  6 


e) 


^^_^KF_  A{l  +  e)^ 

e  e 


For    Hyperbola,    for     same    reason     IC    = 


Also, 
IF 


A{e—1) 


;  and  KC 


KF 


^(e+1) 


e  e 

(5.)    For  Ellipse,    FC  : 


For  Parabola,  see  {4S). 
A{l-e) 


IF+IC=^  —  ^e  + 


^(1  -  es) 


36 


THE   CARTESIAN   METHOD   OF   CO-OEDINATES. 


Ae 


A       A(e^  —  1) 


For  Par- 


For  Hyperbola,    FC=  IF+  lO  =Ae—A4- 

e  e 

dbola,  FC  ^  G  F  =  p,  by  definition. 

(6.)  For  Ellipse,  D  being  a  point  in  the  curve  wliose  distance  from  the   direc- 
trix is  OC,  we  have  OC  = =  — . 

e  e 


For  hyperbola,  OC  =  Ol  —  CI 


A 


Ae 


=  — .     (The  distance  in  the  ellipse  may  be  obtained  in  the  same 


way.)     For  Parabola,  same  conception  as  in  (1)  above. 

(7.)  For  Ellipse  and  Hyperbola  see  Ex's.  7  and  12.     For  Parabola,  1  —  e'^  =  l  —  1 

=  0.  ■ 

(8, )  For  Ellipse,  GF  =p=  FC  X  e  =  A{l  — e'^)  = -j-.      For   Hyperbola,    G  F 


A 


jB2 


=  p=FCXe  =  A{e^  —  1)  =  — .     For  Parabola,  G  F  =p  by  definition. 


SO,  ^Toh,  To  pass  a  conic  section  through  three  given  points,  so 
that  it  shall  have  a  given  focus  ;  and  to  determine  its  elements  ;  i.  e.,  the 
axes,  foci,  directrix,  eccentricity,  etc.,  if  an  ellipse  or  hyperbola,  or 
the  latus  rectum  if  a  parabola. 


Solution. — Let  M,  N,  and  O,  Fig.  38,  be  the 
given  points,  and  F  the  given  focus.  Connect  the 
points  with  the  focus,  and  draw  O  N ,  and  N  M , 
and  produce  them  towards  the  probable  position 
of  the  directrix,  as  to  L  and  K.  Now,  take  a 
point  R,  on  QL,  such  that  OF  :  N  F  :  :  OR  : 
NR,*  and  R  is  a  point  in  the  directrix.  In  like 
manner,  take  NF:MF::NS:MS,  and  S  is 
another  point  in  the  directrix.  Hence  the  direc- 
trix can  be  drawn. 

To  prove  that  a  line  drawn  through  R,  and  S, 
as  AB,  is  the  directrix,  we  have  to  show  that 
O^F  _  N  F  _  MF 
OP  ~ 


EiG.  38. 


OP,     NQ,    and    MX, 


NQ         M 

being  perpendicular  to  AB.     Now  OP:  NQ  ::OR  :  NR.     But  by  construc- 

O^F  _    N^ 
p6  "~ 
NF 


ton  OR:NR::OF:NF. 


In  like  manner  NQ  :  M" 


.-.   OP  :  OF  ::  NQ  :  NF,    or 
N  S  :  M  S  :  :  N  F  :  M  F. 


NQ 
MF 


NQ  MT 

Q.  E.   D. 

To  make  the  numerical  computations  requires  much  more  labor  than  to  effect 
the  geometrical  solution.  We  may  proceed  as  foUows  :  Having  the  distances  O  N , 
NM,  OF,  N  F,  and  MF  given  in  numbers,  compute  the  numerical  values  of 
N  R  and  M  S  from  the  proportions  used  in  the  construction.  The  sides  of  the 
triangles   O  F  N    and    N  F  M   being  known,  their  angles  can  be  found  by  trigo- 


*  This  coustruction  is  effected  tbiis:  taking  tlio  proportion  by  division,  (OF  —  N  F),  or  OG  :  O  F 
:  :  ( O  R  —  N  P. ),  or  O  N  :  O  R .  From  this  proportion  O  R  can  be  constructed,  a.s  the  other  terms  are 
known. 


OF  THE  CONIC  SECTIONS. 


37 


nometry;  whence  we  get  the  angle  R  N  S,  as  it  equals  180° —  (O  N  F  -{-  FN  M). 
Then,  in  the  triangle  RNS,  we  shall  have  two  sides  and  the  included  angle  ; 
whence  the  angle  N  R  S  can  be  found,  and  from  it  N  R  Q  becomes  known.  Now, 
in  the  right  angled  triangle  N  RQ,  we  know  the  hypothenuse  and  one  acute  angle, 
and  can  find  N  Q.  Again,  letting  fall  the  perpendicula.r  N  E,  forming  the  trian- 
gle FNE,  we  can  compute  EF,  since  FN  is  known  and  the  angle  FNE  = 
FNR  +  RNQ—  90°.      FC  is  therefore  known,  being  equal  to  EF  -f  NQ. 


As 


NF 


NQ 

tricity,  is  known 


is  now  determined,   the  ratio  e,  or,  what  is  the  same   thing,   the  eccen 

FH  NF 


Taking  a  point,  as  H,  upon  FC,  such  that 


the 


HC  NQ 

vertex  is  determined.  In  a  similar  manner  the  other  vertex  of  an  ellipse  or  hyper- 
bola can  be  found.  Letting  p  be  half  the  latus  rectum,  F  U ,  it  can  be  found  from 
FN  p 


NQ 


FC 


Ex.  1.  Construct  a  conic  section  passing  tlirough.  the  points  O,  M, 
and  N,  and  having  F  for  a  focus,  knowing  that  O  F  =6^,  N  F  =3^, 

M  F  =  21,  O  N  =  v/18,  N  M  =  v/10.     Let  the  geometrical  con- 
struction be  given,  and  also  the  numerical  solution. 
The  locus  is  a  parabola  whose  latus  rectum  is  9. 

Ex.  2.     Construct  and  compute  as  above,  when  OF  =2.08,  N  F 
1.08,  M  F  =  .46,  ON  =  1.12, and  N  M  =  .87. 

Ex.   3.  Construct  and  compute  as  above,  when  O  F  =  10,   N  F  "= 
G,  M  F  =  3,  ON  =  6,  and  N  M  -=  4. 


SI,     JPvob,     To  produce  the  general  equation  of  a  Conic  Section 
referred  to  rectangular  axes. 


Y 

M 

/" 

c 

\           \        V 

/ 

1 

N 

\ 

\ 

A 

/S  D      K 

\^^ 

X 

/z' 

\B 

Fig.  39. 


Solution. — Let  MN,  Fig.  39,  be  an  arc  of 
any  conic  section,  F  the  focus,  C  B  the  direc- 
trix, ZZ'  the  axis  of  the  curve,  and  AX  and 
AY  the  axes  of  reference.  Let  P  bo  v.ny  point 
in  the  curve,  and  its  ordinate  P  D  :  also  draw 
the  ordinate  of  the  focus,  FK.  Draw  from 
the  origin  AG  perpendicular  to  CB.  Draw 
DH  parallel,  and  P L  perpendicular  to  CB. 
X   Join  P  and  F,  and  draw  PI  parallel  to  AX. 

Let  AG,  the  distance  from  the  origin  to  the 
directrix,  be  represented  by  d  ;  the  co-ordinates 


of  the  focTis,  A  K  and  F  K,  by  m  and  n  respectively  ;  the  ratio  mentioned  irt  the 


definition  {4i2), 


P  F 


PE 

axis   of  abscissas,    ZSX  =  GAX 
nates,  A  D  and  PD,  by  a  and  y. 


,  by  e  ;  the  angle  which  the  axis  of  the  curve  makes  with  the 
LDP,    by  a  ;    and  the  general  co-ordi- 


38 


THE   CARTESIAN   METHOD   OF   CO-ORDINATES. 


Now,   PF2  =e2  .  pE-^      But  PF-  =  PI-  +  Fl'^  —  (m  —  a;)2  -f  {n  —  yY. 
Again,    PE2  =  (PL  +  AH  —  AG)-2  =  (PD  sin  a  -\-  AD  cos  a  —  dy  = 

{y  sin  a  -f-  ^  ^^^  ^  —  ^^y'-     Whence,  substituting  we  have 

(Eq.  A),     (m  —  x)-  -{•  {n  —  yy^  =  e'{y  sin  a  -j-  x  cos  a  —  d)^. 

S2.     Cor.  1. — The  equation  of  the  ellipse  and  hyperbola  re/erred  to 
their  axes  is 

2/2  +  (1  —  e^')  x^  =  A'-{\  —  e=), 
in  ichich  A  is  the  semi-transverse  axis,  and  e  the  eccentricity,  or  the  char- 
acteristic ratio  {4:2). 

Dem. — Let  the  curve  in  Fig.  39  be  conceived 
to  change  position  so  as  to  assume  that  in  Fig. 
40  or  41,  the  axis  of  the  curve  coinciding  with 
the  axis  of  abscissas,  the  origin  at  the  centre, 
A,  F  the  focus  and  CB  the  directrix.  As 
the  axis  of  the  curve  now  coincides  with 
the  axis  of  abscissas,  <x  =  0.  As  the 
focus  falls  upon  the  axis  of  abscissas,  n  =  0. 
By  consulting  (4:9)  it  will  be  seen  that 
m  =  =F  A  F  =  q=  Ae,  and  d  =  '=f  AG  =  ^ 


Y^ 


Fig.  41. 


e* 
Fq. 


Yf hence,   substituting  these  values  in 


A,   we   have  (  =f:  Ae 
A 


(A  \- 
x  rb  —  )   =    (ex  ± 


A)^. 


Expanding 


and  reducing,  we  have 

{Eq.    B),  2/2  +  (1  —  e2)a;2  =  ^^(i  _  e-). 

Q.  E.  D. 

QuEET. — How  is  it  that  the  same  equa- 
tion is  made  to  represent  two  curves  so 
different  from  each  other? 


dSo  CoE.  2.— The  equation  of  the  ellipse  re/erred  to  its  axes,  and  in 
terms  of  its  semi-axes  is 

in  which  A  and  B  are  the  semi-axes. 


*  The  upper  sign  applies  to  the  ellipse,  and  the  lower  to  the  hyperbola. 

The  same  final  result  may  be  obtained  in  the  case  of  the  hyperbola  by  conceiving  the  axis  ZZ' 
of  Fig.  30  to  have  revolved  to  the  left  and  the  branch  to  fall  on  the  left  of  the  origin.  In  this 
case  a  =  180°,  cos  a  =  —  1,  and  x  is  negative. 


OP  THE   CONIC   SECTIONS. 


39 


Dem. — This  equation  is  obtained  at  once  from  Eq.  B  hy  substituting  for  1  —  e^ 
its  value,  -—  (^9),  and  clearing  of  fractions. 

S4:,  Cor.  3. — The  equation  of  the  hyperbola  referred  to  its  axes,  and 
in  the  terms  of  its  semi-axes,  is 

A^y^  —  B^x^  =  —  A'^B% 
in  which  A  and  B  are  the  semi-axes. 

B^ 
Dem. — Substituting  in  Eq.  B, —  for  1  —  e^  (49),  we  have,  after  clearing  of 

fractions,  A^y^  —  B'^x^  =  —  A^BK 
ScH. — These  two  equations,  viz., 

^iyi  _|_  B-^x"^  =  A^B%  for  the  ellipse,  and 
^2y2  _  _B2  x^  =  —  AW%  for  the  Hyperbola, 

are  the  most  important  forms  of  the  equations  of  these  loci,  and  are  always 

meant  when  their  equations  are  spoken  of  without  specification.     They  may 
be  called  the  Common  Forms. 


SS»  CoK.  4. — Tlie  equation  of  the  ellipse  referred  to  its  transverse  axis 
and  a  tangent  at  ihe  left  hand  vertex,  is 

B\ 


and  of  the  hyperbola, 


B^ 


Dem. — In  either  of  the  annexed  figures, 

let   AX    and  AY   be   the  axes,    F   the 

focus,  and  C  B  the  directrix.    Now  a  =  0  ; 

n=0;   m=^A^Ae;   d=  —  AG=  — 

A — Ae       Ae — A    .      ,.        „. 

= m    the    ellipse,    and 

e  e 

4"  AG  = — —  in  the  hyperbola,  hence 

e 

in  general  d  = (49),  Substitut- 
ing these  values  in  Eq.  A  (SI),  and  reduc- 
ing, we  have 

2/2  +  (1  —  e2)ic2  _  2^(1  _  e^)x  =  0, 
which  is  appHcable  to  either  locus.     For 

B^ 

1  —  e2  substituting —-  for  the  ellipse,   and 
A^ 

B^ 
—  —  for  the  hyperbola,  we  have 

J52 


Fig.  42. 


y2  =  -77(^Ax  —  a;2),  and 
B^ 

y*  =  —  ^  (2<5fa5  — «2).       Q.  B.  D. 


40 


THE   CARTESIAJN'    METHOD   OF   CO-OKDINATES. 


SG»  Cor.  6. — The  equation  of  the  circle  referred  to  a  pair  of  diameters 
at  right  angles  to  each  other  is 

2/2  -f  572  =  B2,  THE  Common  Form. 
When  the  reference  is  to  a  diameter  and  a  tangent  at  its  left  hand  extremity, 
the  equation  of  the  circle  is 

y^  =  ^Rx  —  x^.  Y^  Y 

Dem.  — These  two  forms  are  readily  produced  from 

A^y^  -f  ^2  a;2  =  A^B^  and  y^-  =  -j-A2Ax  —  x^),    by 

making  A  =  B  =  R,  the  radius  of  the  circle,  and 
dividing  out  the  common  factor  B^.  For  the  first 
form  the  origin  is  at  A,  Fig.  43,  and  for  the  second 
form  at  A'. 

S7»  Cor.  6. — Making  K  =  'B  in  the  equations  of  the  hyperbola^  Cors. 

3rd  and  4th,  there  residt  j^  ■ —  x^  ^  — A^,  and  y^  =  — (2 Ax  —  x^),  as 
equations  of  the  Equilateral  Hyperbola  (47). 

S8,  Cor.  7. — To  obtain  the  equation  of  the  hyperbola  conjugate  to  A^y- 
—  B^x^  =  —  A'^B"',  and  referred  to  the  same  axes,  we  have  but  to  change 
the  sign  of  the  absolute  term,  and  write  Asys  —  B^x^  =  A^B^. 


Dem. — Conceive  the  curve  Fig.  39,  to  take 
the  position  M  N ,  Fig.  44,  the  axis  of  the 
curve  and  the  focus  falling  on  the  axis  of 
ordinates,  and  the  origin  being  at  the  cen- 
tre,   A.     We  then  have  a  =  90°,  m  =  0, 

B 
n  =  AF   =  Be,  and  d  =  AG    =  — • 

Whence,  substituting  in  Eq.  A,  it  becomes 

/  7?  \2 

a;2  4_  (5e  —  yy^  =  e'-  [y  —  —  j  =(e2/— -B)2  ; 

or,  expanding  and  reducing,  x^  -{-  {1  —  e^) 
y2  ==    B^    (1  —   e2).      But   1  —  e2  =   — 

— ,  which  substituted,  gives  after  reduction,  A^^  —  B^x^  =  A'^BK    q.  e. 


D. 


so.  Cor.  8. — The  equation  of  the  Parabola  referred  to  its  axis  and 
a  tangent  at  the  vertex,  is  y^  =  2i^x,  in  which  2p  is  the  latus  rectum. 


OF  THE  CONIC   SECTIONS. 


41 


c 

Y 

M 

X'        G 

B 

A 

H 
Y' 

Dem. — Besuming  Eq.  A,  and  conceiving  the 
curve  situated  as  in  Fig.  45,  a  =0,  n  =  0,  m 
=  AF  =  ip,  d  =^  —  AG  =  —  ^p,  and  e 
=  1.  Substituting  these  values,  we  have 
(^p  —  ic)2  4-  2/2  =  (cc  +  ^)2  ;  or,  reducing, 
2/2  =:  2px.  Q.  E.  D.  This  is  the  Common  Equa- 
tion of  the  parabola. 


Fig.  45. 

SO,  ScH. — It  will  be  observed  that  the  difference  in  form  between  the 
equation  of  the  ellipse  in  terms  of  the  semi-axes,  and  the  corresponding 
equation  of  the  hyperbola  may  be  considered  as  embraced  in  the  sign  of  B^ ; 
so  that  hereafter,  in  any  case  when  a  property  of  the  ellipse  is  deduced 
from  the  equation  of  that  locus — as  many  Avill  be, — the  corresponding 
property  of  the  hyperbola  can  be  discovered  by  simply  substituting  —  jB2 

for  B^  in  the  result ;  or  if  5  only  is  involved,  by  replacing  it  hj  B\^  —  1. 
Our  subsequent  work  may  often  be  much  abridged  by  this  means.  It  is 
also  to  be  remarked  that,  if  the  formula  expressing  any  property  of  either 
locus  does  not  contain  B,  i.  e.,  does  not  depend  upon  the  conjugate  axis, 
such  property  is  the  same  in  both  loci. 


01,  I*TOp»  Every  equation  of  the  second  degree,  between  two  vari- 
ables, is  an  equation  of  a  conic  section. 

Dem. — Kesuming  Eq.  A,  expanding  and  collecting  terms,  we  have/ 

(1  —  e2sin'^«:)2/''  —  2e'-sin  acos  axy  -{-(I  —  e^co^"a)x~  -\-{2e''d  sin  a  —  2n)2/ 

4-(2e-d  cos  a  —  2m)a;  4-(m2  -f-  n"^  — e-d2)=  0.     {Eq.  B.) 
Representing  these  coefficients  in  order  by  A,  B,  C,  etc. ,  we  have — 
Ay'^  -f  Bxy  ^  Cx^ -{- By  ^  Ex -^  F  =  0.     (Eq.  A'. ) 

This  is  the  Complete  Equation  of  the  second  degree  between  two  variables  ;  i.  e.  it 
contains  every  variety  of  terms,  with  respect  to  the  variables,  which  such  an  equa- 
tion can  have. 

It  now  remains  to  be  shoAvn  that  these  coefficients  may  have  such  values  (by  the 
locus  being  of  a  proper  species  and  properly  situated)  as  to  cause  the  equation  to 
take  any  and  every  given  form.  Dividing  Eq.  A'  by  F,  (any  one  of  the  coefficients 
A,  B,  G,  etc. ,  would  do  as  well)  and  distinguishing  the  resulting  coefficients  by 
accents,  we  have  A'y^  +  S'xy  -f-  C'x-  -j-  D'y  -f-  Ex  -f-  1  =  0.  Now  the  five  coef- 
ficients, A',  B',  C,  D' ,  E,  depend  upon  the yiue  arbitrary  constants,  a,  m,  n,  d,  and  e, 
in  such  a  way  that  such  values  may  be  assigned  to  these  last  quantities,  i.  e.,  the 
locus  may  be  of  such  species  and  so  situated,  as  to  give  the  quantities  A',  B' ,  C', 
D',  E,  severally  any  and  all  required  values.  Hence  every  equation  of  the  second 
degree  between  two  variables  is  an  equation  of  a  conic  section,     q.  e.  d. 

[Note. — If  the  above  demonstration  seems  abstract,  let  the  student  consider  a 
special  case.     For  example,  let  us  inquire  if  2y2  —  3xy  -\-2y  —  5.r  -f-  4  =  0,  is  an 


42  THE  CARTESIAN  METHOD  OF  CO-ORDINATES. 

equation  of  a  conic  section  ;  and,  if  it  is,  of  what  species  is  the  locus,  and  how 

situated.     Now,  dividing  this  equation  through  by  4,  we  have  ^y^  —  ficy  -j-  jy  — 

1^  _)_  1  =  0.     Comparing  this  with  the  equation    A'y^  -{-  B'xy  -\-  Cx^  -f-  D'y-\- 

.     .      ,,    ,    .,        1  —  e-s,m- a      ^,       —  26-  sin «  cos  a: 
E'x  4-1=0,  and rememDenng that  A  =— — - — ; 7-,  is  = ; ■- — , 

1  —  e2  cos2  a      ^,       2e'd  sin  a  —  2n          ,     _,,        2e^d  cos  a  —  2m 
C  = T-,  D  = : ;— )    and    E  =  ; --r— ,  we  can 

write  the/iue  following  equations  : 
1  —  e2  sin2  a         1 


(2). 


—  2e2  sin  a  cos  a: 3 

yn,2  -j-  n2  _  e2(^2  4 


(3). =  0,  the  coefficient  of  a;^  in  this  special  example  being  0  ; 

^  ^    m2  +  ?i2— e2(Z2  i-  r  &     . 


(4). 
(5). 


2e2d  sin  a  —  2n 1  _ 

m2  4-  n2  —  e2d2"  ""  2  ' 
2e2d  cos  a  —  2m  5 


?n2  -|-  rfi  —  e2d2  4 

Now,  from  these  five  equations,  the  values  of  the  jive  quantities  a,  m,  n,  e,  and  d, 
can  be  found.  But  these  being  known,  the  species  (determined  by  e)  and  the 
situation  of  the  locus  are  known.  As  a  similar  course  could  be  pursued  with  any 
particular  equation  of  the  second  degree  between  two  variables,  it  is  evident  that 
every  such  equation  represents  some  conic  section.] 


02*  JPvoh,  To  determine  the  features  of  an  equation  of  the  necond 
degree  between  two  variables,  which  characterize  the  several  species  of  conic 
sections. 

Solution. — Comparing  Eq.  A'  with  Eq.  B,  we  see  that  A=l  —  e^  sin2  a,  B  = 
—  2e-  sin  a  cos  a,  and  0=1  —  e^  cos^a.     Squaring  the  value  of  B,  and  subtract- 
ing from  this  square  4  times  the  product  of  A  and  C,  we  have 
^2 _^  4^ C=  4e4  sin2  a  cos2  a  —  4(1  —  e2  sin^  a)  (1  —  e^  cos2  a) 

=  4e^  sin2  a  cos2  a  — 4(1  —  e^  sin2  a  —  e^  cos2  a-\-  e'^  sm^a  cos2  a) 
=  4e4  sin2  a  cos2  a  —  4  -f-  4e2(sin2  a  +  cos^  a)  —  46^  sin2  a  cos^  a 
=  —  4  -{-  4e2,  since  sin2  cc  -\-  cos^  a  =  l. 
...  52 _4^ (7  =  4(62 _i). 

Now,  in  the  parabola,  e  =  1  ;  in  the  eUipse,  e  <<  1  ;  and  in  the  hyperbola,  e  >>  1. 

Therefore, 

52 — 4J.(7=  0  characterizes  the  Parabola  ;  ^ 

B^  —  4:A  G<C  0  characterizes  the  Ellipse  ;       V    (2)). 

B'^  —  4AC>  0  characterizes  the  Hyperbola.  J 

Ex.  1.  Determine  the  species  of  the  locus  of  the  equation  2?/2  — 
Zxy  +  5^2  _  2i/  —  12  =  0. 

SuG.— As  the  equation  is  of  the  second  degree  the  locus  is  a  conic  section. 
Again,  in  this  case,  ^  =  2,  ^  =  —  3,  and  6'=  5.  .-.  B^  —  4A(7=  9  —  40  =  — 
31  <!^  0  ;  and  the  locus  is  an  eUipse. 

Ex.  2.  Determine  the  species  of  the  locus  of  the  equation  y"^  — 
^xy  —  3a?2  +  2^7  —  8  =  0.  The  locus  is  an  Hyperbola. 


OF  THE  CONIC  SECTIONS.  43 

Ex*s  3  to  10.  Determine  the  species  of  the  loci  of  the  following 
equations  :  (1)  Sy'-  —  2x^  —  4?/  +  20  =  0.  (2)  4?/2  —  2y  -\-  x=0. 
(3)  4:xy  =  16.  (4)  2y  — 3^  =  4(^  —  6)=+ (^  +  2/)^.  (5)  d{x—yy 
=  2{a;  +  3).     (6)  y^-  =  3{x  —  2).      (7)  y^  —  5x  =  2{x  —  yy  +  y, 

The  1st  is  an  Hyperbola ;  the  2nd,  a  Parabola;  the  3rd,  an  Hyper- 
bola ;  the  Uh,  an  Ellipse ;  the  5th,  a  Parabola ;  the  6th,  a  Parabola ;  the 
1th,  an  Hyperbola. 

Sug's.-  Equations  like  the  4th  must  be  put  in  the  form  Ay-  -f-  Bxy  -j-  Ox"^  + 
Dy  4"  Ea  -j-  F=0,  before  applying  the  test.  Thus  equation  (4)  becomes  y^  -f~ 
2xy  +  5x2  —  2y  —  Six  +  100  =  0.  From  this  ^  =:  1,  B  =  2,  and  6'  =  5.  .  • . 
B2  _  4:AC=~  16  <  0.     In  (3),  ^  =  0,  ^  =  4,  and  G=Q. 

OS*  Cor.  1. —  The  species  of  the  locus  depends  solely  upon  the  coeffi- 
cients A,  B,  and  C. 

64.  CoR.  2. — The  form  Ay'  +  Cx^  +  Dy  +  Ex  +  F  =  0  embraces 
all  species  and  varieties  of  the  conic  section. 

Dem.  — Whatever  the  locus  may  be,  if  the  axis  of  abscissas  is  assumed  parallel 
to  the  axis  of  the  locus,  a  =  0.  Hence  sin  a  =  0  ;  and  B,  which  equals 
— 2e2  sin  a  cos  a,  is  0.  In  like  manner,  by  assuming  the  axis  of  abscissas  perpen- 
dicular to  the  axis  of  the  locus,  cos  a  =  0  ;  and,  consequently,  B  ^^  0.     q.  e.  d. 


OS,     JPf*oh,     To  determine  the  varieties  of  the  ellipse. 

Solution. — As  the  equation  Ay-  -j-  Cx-  -f-  Dy  -\-  Ex  -\-  F=0  includes  all  spe- 
cies and  varieties  of  the  conic  sections  {64),  we  have  only  to  consider  what  loci  it 
represents  when  the  condition  B'^  —  44(7<  0  is  fulfilled.  This  condition  can  be 
fulfilled  only  when  A  and  Oliave  like  signs  and  are  both  numerically  greater  than  0; 
for,  as  jB  =  0,  if  -4  and  C  have  different  signs,  B^  —  4JL6',  becomes  -f-  4: AC,  which 
is  greater  than  0.  If  J.  or  C  =  0,  J5-  —  4  J^C  =  0.  Now,  as  the  signs  of  A  and  C 
must  be  ahke,  we  miay  consider  them  as  always  -f-  ;  because,  if  they  were  —  in  a 
given  case,  the  signs  of  all  the  terms  of  the  equation  could  be  changed.  Again, 
since  D  and  E  depend  upon  m  and  n  (D  =  2e"d  sin  a  —  2n,  and  E  =  2e-d  cos  a  —  2m) , 
such  values  may  be  given  to  m  and  n,  i.  e.,  the  origin  may  be  so  located  with 
reference  to  the  focus,  that  B  and  E  shaU  each  be  0  ;  nor  does  this  affect  the  spe- 
cies of  the  locus,  as  A  and  C  do  not  depend  upon  m  or  n.  Hence  we  learn  that  the 
form  Ay'^  +  Cx^  -{-  F=0,  embraces  all  the  varieties  of  the  ellipse. 

On  this  form  we  observe  that  if  ii^is  negatiif€*Jt  gives  by  transposition  Ay-  +* 

Ck^=sF,  or  tKe  equation^bf  the  ellipse  refei^"^  to  its  axes,  in  which       1-    and. 

I  —  are  the  semi-axes  of  the  curve.     If  A  and   C  are  numerically  unequal  the  • 

axes  are  unequal,  and  we  have  the  Common  Ellipse.     If  A=  C,  the  axes  become, 
equal  and  the  locus  is  a  Circle.      Again,   if   F=  0,  Ay^  -f  Cx'^  =  0,  gives  y  =  ±: 

-  x\  which  gives  no  real  values  except  x=  0,  y  =  0  ;  and  hence  represents 

a  PoirU  (the  origin).     Finally,  if  iS^is  +,  in  the  form  A^^  -f  CScs  -f.  F^a  0,  we  have 


4 


44  THE   CARTESIAN   METHOD    OF   CO-ORDINATES. 


?/  =  ±      / — —,  in  wMcli  all  real  values  of  one  variable  give  imaginary  values 

to  the  other  ;  hence  the  equation  has  no  locus  in  the  plane  under  consideration. 

There  are,  therefore,  4  varieties  of  loci  embraced  in  the  equation  of  the  second 
degree  between  two  variables,  which  fulfill  the  condition  B^ —  4J.C<^0,  and  are 
hence  called  varieties  of  the  ellipse  ;  viz.,  the  Ellipse  proper,  the  Circle,  the  Point, 
and  the  Imaginary  locus. 


00 »  JPtoI),   To  determine  the  varieties  of  the  hyperbola. 

Solution. — Resuming  the  equation  Ay- -\- Cx--\- Dy -{- Ex-{- F=0,  which  in- 
cludes all  siDecies  and  varieties  of  conic  sections  (6^),  we  observe  that  the 
characteristic  condition  of  the  hyperbola,  B-  —  4:AC^0,  can  be  fulfilled  only 
when  A  and  C  have  opposite  signs,  and  are  both  numerically  greater  than  0, 
since  B  =  0.  Also  that,  as  i>  and  E  depend  upon  m  and  n,  and  A  and  C  do  not, 
such  values  may  be  given  to  m  and  n,  i.  e.,  the  origin  may  be  so  situated  with 
respect  to  the  focus,  that  D  and  E  shall  each  be  0.  Hence  Ay-  —  Cx-  -j-  i^=  0, 
embraces  all  the  varieties  of  the  hyperbola. 

On  this  form  we  observe  that  if  i^is  positive,  it  gives  by  transposition  Ay'^  —  Ox^ 

If, 

=  —  ^  or  the  equation  of  the  hyperbola  referred  to  its  axes,  in  which       I -r,  is  the 

l-~F 
semi- transverse,  and       I is  the  semi-conjugate  axis.     If  A  and  Care  numeri- 

\ 
cally  unequal  these  axes  are  unequal,  and  we  have  the  common  form  of  the  hyper- 
bola.    If  A  =  C,  the  axes  become  equal  and  the  locus  is  an  Equilateral  Hyperbola. 
Again,  if  i<^is  negative  the  equation  becomes  Ay- —  Cx-  =  F,  which  is  the  equation 
of  the  y  hyperbola,  since  the  real  axis  is  on  the  axis  of  y ",  and  the  imaginary  one 

IX' 

on  the  axis  of  £C.*     Finally,   if  1^  =  0,   we  have  vl?/'  —  Cx- =  0,  or  y=d:z      l-x, 
which  is  the  equation  of  two  straight  lines  passing  through  the  origin  and  making 

r^'       re 

angles  with  the  axis  of  x,  whose  tangents  are  respectively      |  -^  and  —     |  — . 

There  are,  therefore,  3  varieties  of  loci  embraced  in  the  equation  of  the  second 
degree  between  two  variables,  which  fulfill  the  condition  B-  —  4cAC  '^0,  and  are 
hence  called  varieties  of  the  hyperbola  ;  viz.,  the  Hyperbola  with  unequal  axes, 
both  on  the  axis  of  x,  and  on  the  axis  of  y,  the  Equilateral  Hyperbola,  and  Two  Bight 
Lines  intersecting  each  other. 


07*  I^Toh,   To  determine  the  varieties  of  the  parabola. 

'3oiiUTioN.  — As  the  equation  Ay-  -f-  G^'  +  ^If  ~\~^^-\~  ^=  ^j  embraces  all  species 
and  varieties  of  the  conic  sections,  we  have  to  determine  only  what  loci  it  repre- 
sents when  B^  —  4J.C=0.     But  as  B  =  0,  this  condition  can  be  fulfilled  only  by 


*  This  form  of  expression  is  frequently  used  instead  of  "  axis  of  ordinates,"  and  "  axis  of  abscis- 
sas." 


OF   THE   CONIC    SECTIONS.  45 

A  =  0,  or  G=0.    Now  as  the  equation  is  symmetrical  with  respect  to  cc  and  y,  it  will 

be  sufficient  to  examine  the  case  in  which  C  =  0,  or  the  form  Ay'-  -{-  Dij  -{-  JEx  -\- 

F=  0.    Kemembering  that  a  has  been  made  0,  and  that  e  =  l,  we  find  D  (which 

equals  2e'd  sin  a  —  2)i)  =  —  2n.     This  can  now  be  made  0  by  taking  the  axis  of 

the  curve  for  the  axis  of  x,  and  the  equation  takes  the  form  Ay^  -\-  Ex  -[-  F=.Q. 

But  in  this  case  we  cannot  make  E  =  (2e-cZ  cos  a  —  2m  =  2tZ  —  2m)  =  0,  since 

that  would  require  that  d  =  m,  which  is  absurd,  since  d  is  the  distance  from  the 

origin  to  the   directrix,  and  m  is   the  distance  from  the   origin  to  the  focus. 

{Numerically  d  may  equal  m  ;  but  E  =:  0  requires  that  they  also  have  the  same 

•sign). 

We  therefore  have  to  discuss  the  equation  Ay"^  -f-  -^^  "f"  ^=  ^'  which  includes 

all  varieties  of  the  parabola.     As  F  depends  upon  m~  -\-  n'^  —  e-d^,  and  as  n  =  0, 

and  e  =  1,  jPmay  be  made  0,  by  putting  m  =  —  d,  which  requires  only  that  the 

E 
origin  be  at  the  vertex.     The  equation  is  thus  reduced  to  y"  =  ±z  —oc,  the  dr  sign 

being  given  to  E,  as  no  restriction  has  been  imposed  upon  it.   This  is  the  common 

E 
equation  of  the  parabola,  in  which  —  =  2p.     The  -\-  sign  locates  the  curve  at  the 

right  of  the  origin,  and  the  —  sign  at  the  left,  but  both  give  the  same  variety.  Again, 
if  in  Ay^  +  Bxy  +  Cx^  _^  x>y  +  £r  +  F=  0,  we  make  ^  =  0,  JS  =  0,  and  C  =  0, 
the  condition  B-  —  4  J.  C  =  0  is  fulfilled,  and  the  locus  is  therefore  sometimes  called 
a  variety  of  the  parabola.  This  locus  is  evidently  a  right  line,  its  equation,  Dy  -j- 
Ex  -f-  F=^  0,  being  an  equation  of  the  first  degree  between  two  variables.  *  Finally, 
if  an  equation  of  the  second  degree  between  the  two  variables  can  be  reduced  to 
either  of  the  forms  ?/--[-  '2xy-\-x-  zb  P{x  -\-  y)-\-S  =  0,  y"  —  2xy  -{-  x-  ±  F'  {y  —  x) 
4-  S'  =  0,  or  t/2  ±  2xy-}-x-^  -\-  S"  =0,  the  condition  B^  —  4.AG=0is  still  fulfilled, 

although  the  equation  may  bo  reduced  to  the  form  y  dzX  =  m  ±  Vp  —  q,  which 
is  the  equation  of  two  real  or  imaginary  parallel  right  lines, 

"VVe  have,  therefore,  4  varieties  of  the  Parabola  ;  viz.,  the  Common  Parahola,  the 
Bight  Line,  Two  ParUllel  Bight  Lines,  and  Two  Parallel,  Imaginary  Bight  Lines. 


OS*  Cob.  1. — The  eccentricity  of  the  circle  is  0,  and  the  directrix  is 
at  infinity. 

Dem. — In  obtaining  the  equation  of  the  circle  (OS),  we  made  a  =  0,  and  A=  C. 
Hence  1  =  1  —  e'~,  or  e  =  0.  Again,  when  the  ellipse  passes  into  the  circle,  the  foci 
unite  in  the  centre.     Now,  calling  the  distance  from  any  point  in  the  curve  to  the 

directrix  s,  and  the  radius  of  the  circle  B,  we  have  —  =  0  (the  distance  from  any 

point  in  the  curve  to  the  focus  divided  by  its  distance  from  the  directrix  equals 
the  eccentricity)  ;  whence  s  =  (X).+ 


*  In  reality  this  condition  is  not  compatible  with  our  fundamental  hypothesis,  which  requires 
the  equation  to  be  of  the  second  degree.  Moreover,  the  conditions  A  =  0,  B  —  0,  and  C  =  0,  are 
inconsistent  with  the  character  of  the  coefficients  A,  B,  and  C,  inasmuch  as  they  require   that 

1  _  e^sin^a:  =  0,  1  —  e^cos^a  =  0,  and  2e-sinti:cosa:  =  0,  or  three  arbitrary  conditions  while  there  are 

but  two  arbitrary  constants,  e  and  OC, 

t  If  ^  =  0,  the  couditiou  —  =  0,  is  satisfied  by  any  value  of  s.    In  this  case  the  locus  is  a 


46 


THE  CAETESIAN  METHOD  OF  CO-OEDINATES. 


Ex.   1.  Determine  the  species  and  situation  of  the  locus  i/«  +  Gy 
—  1207  +  33  =  0. 


Solution.  — To  determine  the  species  of  the  locus,  observe  that  the  equation  is 

of  the  second  degree  between  two  variables,  and  fulfills  the  condition  B- 4:A0 

=  0.     Therefore,  the  locus  is  a  parabola. 

To  determine  the  situation  of  the  locus,  compare  its  equation  with  the  general 
equation  of  the  conic  section  :   viz. , 

(1  —  e"^  sin2  a)y'  —  2e2  sin  a  cos  axy-\-{l  —  e^  cos'^  a)x^  -f-  (2e-cZ  sin  a  —  2n)y 
+  (2e''d  cos  a  —  2m)x  -f-  m'^  -}- n-  —  e-d^  =  0. 

As  in  this  example  the  coefficient  of  y'^  is  1,  divide  the  general  equation  through 
by  1  —  e-  sin2  a  before  comparing  the  coefficients.  The  five  equations  from  which 
a,  e,  m,  n,  and  d  are  to  be  found  are — 


(1) 
(2) 
(3) 

(5) 


—  2e2  sm  a  cos  a      ^         „   .  ^ 

— :; ; =  0,  or  e2  sm  a  cos  a  =0. 

1  —  e2  sm2  a 

1  —  e2  cos2  a 


1  —  e2  sin2  a 
2e^d  sin  a  —  2n 


1  —  e2  sin2  a 

2e^d  cos  a  —  2m 
1  —  e^  sin2  a 

m2  -f  ?i2  _  e2d2 


=  0,  or  1  —  e2  cos2  a:  =  0. 
=  6,  or  e-d  sin  <a:  —  n  — 


:3  — 3e2sin2  a. 


=  — 12,  or  e-d  cos  <a:  —  m  =  —  6  -f-  6e2  sin2  a. 


=  33. 


1  —  e2  sin2  a: 

These  equations  are  now  to  be  solved  for  a,  e,  m,  n,  and  d.  But  as  the  locus  is 
a  parabola,  e  =  1.  Also  from  sin  a  cos  a  =  0,  a  must  be  either  0,  or  90°.  From 
(2)  cos  a  =  l.  .-.  a  =  0.  Making  these  substitutions  in  (3),  (4),  and  (5),  they 
become — 

(3 , )  n  =  —  3  ;  (4, )  d  —  m  =  —  6;  (B^)  m^-{-n^  —  d^  =  33.  These  equations  readily 
give  m=5,  and  d  =  —  1 . 

To  construct  the  locus,  let  X'X  and  Y  Y'  be  drawn  at  right  angles  to  each  other. 
Locate  the  focus,  F,  at  (5,  — 3).     As  a  =  0,  draw  the  axis  of  the  locus  parallel  to 


point.    There  are  various  views  which  may  be  taken  of  the  eccentricity  of  a  right  line.    Thus,  consid- 
ering it  as  the  limit  of  the  hyperbola,  we  have  the  equation  y  =        f. 


nI 


Q 

-X,    in   obtaining  which  we  put  F  =  m-  —  e^dr  =  0  (66). 


771  jL  -i4fi" 

From  this  e  =  _.     But  m=  Ae,  and  d=  -.     Hence  e  == = 

d  e  A 


and  c  =  1.      Therefore  y 


-si 


X  is  a  common  limit  of  the 


hyperbola  and  the  parabola.     Again,  if  we  may  be  allowed  to 
consider  Dy  -\-  Ex  -^  F  =  0  [G7)  the  equation  of  a  parabola,  we 
reach  the  same  result  ;  viz.,  «  =  1.    If,  however,  we  consider    »./ 
C  B  the  directrix,  and  a  focus  removed  to  an  infinite  distance,  3 

and   MN  the  Une,    PF,  the  distance  to  the  focus  is  00,  and 

PF 

e  =  -—  =  00.    But  these  speculations  are  curious  rather  than  useful. 
PD 


Tiot.  46. 


OP   THE   CONIC   SECTIONS. 


47 


X'X,  and  through  F.  As  d  =  —  1,  take 
AG  =  —  1,  and  through  G,  drawing  C  B 
perpendicular  to  Z'Z,  it  is  the  direc- 
trix. Through  F  draw  QR  parallel  to 
CB  and  take  QF=  FR  =  FU.  The 
construction  can  now  be  completed  as  in 
(43). 


c 

Y 

A/ 

^^^M 

X'         G 

/ 

X 

Z'      U 

B 

F                                    Z 

Fig.  47. 


2.  Determine  the  species  and  situation  of  the  locus  y-  +  Sary  -f  8072 
—  4.x  =  0. 


Solution. — Since  5^ —  4:AG=  4  — 12  =  —  8  <<  0,  the  locus  is  an  elhpse.     The 
five  equations  from  which  to  determine  a,  e,  d,  m,  and  n,  are — 

—  2e2sinacosa  1  —  e^  cos^  a 


(5) 


1  —  e^  sin2  a 
'ie^d  since —  2n 


e=*  sin2  a 


_ _..       „  ,  .  ^        ,,,      2e2dcosa  —  2m 

(3)     — i .^^^ —  =  0,  or  e^d sina  —  n  =  0;      (4)     — ^ ^^-^ =  —  4 


1  —  e'^  sin2  a 

w^  4-  n^  —  e^d^ 
1  —  e^  sin2  a: 


1  —  e^  sin=^  <x 


0,  or  m2  -f-  7z2  —  e^d^  =  0. 


"We  observe  that  (1)  and  (2)  contain  only  the  unknown  quantities  e  and  a,  and 
hence  are  sufl&cient  in  themselves  to  determine  these  quantities.  From  (2)  we  get 
after  clearing  of  fractions,  substituting  for  cos^  a,  1  —  sin^  a,  and  reducing,  sin  a 


=  -rr-\/e^  4-  2  ;  whence,  cos  a  =  db  -^\/3e'^ 

It  2e 


2.     The  -\-  sign  alone  is  given  to  the 


value  of  sin  oc,  since,  by  reckoning  the  angle  from  0  to  180°,  we  get  all  possible  incli- 
nations of  the  axis  of  the  locus  to  the  axis  of  reference.  Substituting  these  values  in 
(1)  and  reducing,  we  find  e  =  .91-f-  ;  and  consequently,  sin  a  =  .9239,  and 
cos  a:  =  d=  .3827.  .-.  a==  G7o  30',  or  112°  30'.  To  determine  whether  cos  a  is 
-f-  or  — ,  and  hence  whether  a  is  G7°  30',  or  112°  30',  we  consider  (1).  The 
denominator  of  the  first  member  being  necessarily  -{-,  since  1  ^  e^sin^o:,  the 
numerator  must  be  rendered  -(-,  inasmuch  as  the  second  member  is  -|-  But 
cos  a  is  the  only  factor  in  the  numerator  which  can  become  —  and  thus  render 
the  product  -}-.  .-.  cos  a  =  — .3827;  and  a  =  112°  30'.  Substituting  the 
values  of  e,  sin  a  and  cos  a  in  (3),  (4),  and  (5),  they  become  (Sy)  n  =  .7Q5d  ; 
(4i),  m  =  .586—  .317d;  and  (5,)  7n-^  +  n'^  =  .82ri^.  Substituting  these  values 
of  m  and  n  in  (SJ,  we  find  d  =  .72,  and  —  3.34.  Finally  n  =  .765d  =  .55,  and 
^2.55.     m  =  .586  — .317d  =  .36,  andl.64. 


4:8 


THE  CABTESIAN  METSOD  OP  CO-OEDINATES. 


To  construct  the  figure,  draw 
G  G '  througli  the  origin,  making  an 
angle  of  112^  30'  with  the  axis  of 
X.  Locate  the  foci,  F,  F',  at  (.36,  Q. 
.55),  and  (1.64,  —2.55).  Through 
one  focus,  as  F,  draw  a  line  ZZ'  ^' 
parallel  to  GG',  and  it  will  pass 
through  the  other  focus,  F',  if  the 
work  is  right.  Take  AG  =  .72 
and  AG'  =  —  3.34,  and  draw  CB 
and  CB'  perpendicular  to  ZZ'; 
these  will  be  the  directrices.  Now 
the  ratio  e  =  .91,  the  focus  F,  and 
the  directrix  CB,  being  known, 
the  curve  can  be  constructed  (45). 


Fig.  48. 

3.  Determine  the  character  and  situation  of  the  locus,  2xy  —  x  -{-1 
=  0. 


SoiiUTioN.      JB'^  —  4zAC=  4z.  .  • .  The  locus  in  an  hyperbola.     The  five  equations 
from  which  e,  a,  m,  n,  and  d  are  determined  are — 
1  —  e^  sin2  a 


(1) 


(2) 


(3) 


(4) 


C5) 


m''^  -\-  ri^ —  e^(P 
—  2e2  sin  a  cos  a 
m^ -[- ix^  —  eH^ 
1  —  e2  cos^  a 


=  0,  or  1  —  e^  si]i2  a  =  ^  ', 

=  2,  or  —  e^  sin  a.  cos  a  =  ni^  -\-  n^  —  e^d^ 


m2  -j-  rfl  —  62^2 
2e2d  sin  a  —  2n 
m2  +  ?>  —  e2(^ 
2e5d  cos  a  —  2m 


=  0,  or  1  —  e^  cos2  a=^Q  \ 
"-  =  0,  or  e^d  sin  a  —  ji  =  0  ; 

=  —  1,  or  2e2d  cos  a  —  2m  =  e^d^ 


m'^ 


n2. 


,7i2  _|_  n^  _  e^d'^ 

The  general  equation  is  divided  through  by  in--\-n^ — e^cZ^,  since  in  this  example 
the  absolute  term  is  1. 

From  (1)  we  have,  directly,  sin  a  =  -,  discarding  the  negative  root  for  the  rea- 


1 


1    /- 


d=-ve2  — 1. 
e 


son   given   in  the   preceding   solution.       .  • .    cos  a  =  d=      1 1 

Substituting  this  value  of  cos  a  in  (3),  we  have  e  =  v/ 2  =  1.4142-j-.      Hence, 

cosa  =  i  ,    1-1  =  ±^  \l-\=±       l=±^v/2=  ±.7071+,  and 6t  =  450 

or  135°.     Substituting  the  values  of  e  and  sin  a  in  (4),  we  have  n  ^=  \/~ld  ;  and 


OF  THE  CONIC  SECTIONS. 


49 


by  substitution  in  (2),  we  find 
m  =  zb  1.  In  this  case,  it  will 
be  seen  that  if  we  substitute  the 
-|-  value  of  cos  a  in  (2),  m  be- 
comes imaginary,  but  the  — 
value  gives  m  real.     Therefore, 


cos  a  =  —  -v  2,  and  a 


1350. 

Finally,  substituting  in  (5),  we 

1      —  3      — 

find  d  =  —  tV^^»  ^^<1  tn/-^'  or 

—  .35,    and   1.06;    and,    conse- 

1  3 

quently,  n  =  —  -,  and  -.      (The 

locus  is  situated  as  in  Fig.  49 ; 
but  the  construction  is  so  close- 
ly analogous  to  the  preceding, 
that  the  student  will  have  no 
difficulty  in  effecting  it.) 


4.  Determine  the  character  and  situation  of  the  locus  y^  —  2^^/ 
+  2^2  —  2j7  =  0. 

Results,     e  =  .924,   a  =  58°17',  d=  3.1  and  —  .37,   m  =  1.78  and 
.21,  and  n  =  2.27  and  —  .27. 

S-ca. — In  problems  hke  the  preceding  it  is  not  admissible  to  divide  the  general 
equation  through  by  a  coefficient  corresponding  to  one  which  is  0  in  the  partic- 
ular case,  inasmuch  as  this  process  would  reduce  each  coefficient  to  infinity  or  inde- 
termination. 

Thus  for  Ex.  3,  should  we  put  the  general  equation  in  the  form 


y2 


2e2  sin  a  cos  a       ,1  —  e^  cos^  a      ,  2e^d  sin  a  —  2n 

-^y  4-  :; -^-zr-^—^^  +  -1 :;r-   .,  ^.    y  +  etc.  ; 


1  —  e^  sin'^  oc'^^  '  1  —  e^  sin^  a  "  '  1  —  e^  gin^  ^ 
each  of  the  coefficients,  when  the  application  was  made  to  the  equation  2«2/  —  jc 
-j-  1  =  0,  would  be  infinite  or  indeterminate,  since  in  this  example  1  —  e^  sin^  a 
=  0. 


EXEKCISES. 

[Note. — This  list  of  exercises  is  designed  to  give  the  student  an  opportunity  for  making  an  effort 
to  produce  the  equations  himself.  Nothing  new  is  developed  in  them,-  and  the  student  need  not 
necessarily  tarry  till  he  has  mastered  them  all,  though  by  doing  so  clearness  and  breadth  of  view 
will  be  promoted.  Let  every  one  understand,  however,  that  ability  to  investigate— to  reason  for 
himself— is  the  proper  object  toward  the  attainment  of  which  he  should  strive.] 

1.  To  produce  the  equation  of  the  eUipse  referred  to  its  own  axes 
and  in  terms  of  its  semi-axes,  directly  from  the  definition,  with- 
out first  obtaining  the  general  equation  of  the  conic  section. 


r>o 


THE   CARTESIAN  METHOD   OF   CO-OllDINATES. 


Sug's.     ad  =x,    PTD  =y,    AF  =  Ae, 

PF  

and— —  =e.     PF^  =  e^XPE.^     PF2== 

2/2  +  {Ae  +  xy,    and    PE^  =  { h  ^)  • 

...  2/2  _|-  (1  _  e2)a;2  =  ^2(1  _  e^).     por  1  —  e^ 
substituting    — ,  we   have   A^y'^  -f"  ^2-j;2  =_ 


2.  In  like  manner  produce  the  equation  of  the  ellipse  referred  to 
its  transyerse  axis  and  a  tangent  at  its  left  hand  vertex,  i!  e.,  y^  = 

3.  Show  that  the  equation  of  an  ellipse  referred  to  its  conjugate 

A^ 
axis  and  a  tangent  at  the  upper  vertex  thereof  is  x'^  =  —  -z—i2By  +  y^). 


Sug's.    AD=a^,PD: 


-y,  ^p^=e,a.u6.      C 


p  F2  =  e2  X  P  E2.     Also  P  F^  =  (x  +  Ae^ 
+  (J5  +  y)\  and  PE^  =  ("-  +  x)^ 

\  6  / 


4.  Produce  the  common  equation 
of  the  hyperbola,  A^y^  —  B^x^  =  — 
A^B^,  directly  as  above. 


c 

Y 

y/ 

A 

D 

E 

^^ 

~^"~--^ 

p 

•    f^ 

^ 

^ 

G 

I      F 

0 

I 

7 

B 

"^ 

^— 

Y^ 

Fig.  51. 


5.  In  like  manner  as  above  produce  the  common  equation  of  the 
parabola,  y"^  =  2px, 

6.  Show  that  the  equation  of  an  ellipse  i^A-{y  —  y^y  +  B'^{x  —  x-^)^ 
=  A^B'^,  when  x^^  and  y^  are  the  co-ordinates  of  the  centre,  and  the 
axes  of  reference  are  parallel  to  the  axes  of  the  curve. 

•y 
Sug's.     AL=iCx,   OL=yi,  AD=a5, 

PD  =  2/'   a^^   PF2  =  e2  X  PE2.      Also  h 
PF2=PR2-1-  FR2,  PR=y  —  y,,    FR 

=  x  —  xi  +  Ae,  and  PE  =  PH  —  H  E 

=  .r  —  Xi  -] .     Substituting,    (?/  —  y-^y 

+  ic2  —  2x^x  +  2Aex  +  Xj^  —  2AeXi  -}- 
A^e^  =  e^-x^  —  2e"x-iX  -f  2Aex  -\-  e^-x^^  —  ^ 

Transposing  and  collecting  terms, 


c 

^_ 

P 

E 

//- 

\ 

G 

B 

\  ^ 

0 

y 

y^ 

K 

:        L 

t 

3        ■      X 

Fig.  52. 


OF  THE   CONIC   SECTIONS. 


51 


(y  — 2/i)'  +  (1  — e2)fl?2  — (1  — e2)2a;,a;  +  (1  —  e-^)x,2  =  A%1  —  e'^),  or 

(2/  —  2/i  )^  +  (1  —  e2)(a;  —  Xiy  =  A\l  —  e^).     Putting  —  for  1  —  e%   we  have 

^Hy  —  ^l)^  +  JB%X  —  £Ci)2  =  ^2J52. 

7.  Deduce  from  tlie  general  equation  of  the  conic  section  (SI)  the 
equation  of  a  parabola  whose  parameter  is  2p,  referred  to  rectangular 
axes,  the  axis  of  the  curve  falling  on  the  axis  of  abscissas,  and  the 
vertex  of  the  curve  being  at  ^i  to  the  right  of  the  origin.  Also  the 
equation  when  the  vertex  is  at  x^  to  the  left  of  the  origin.  Also  the 
equation  when  the  axis  is  parallel  to  the  axis  of  x,  and  the  vertex  is 
at  {x,,  y^). 

The  equations  are  y^  =  2p(^  —  x^),  y^  =  '2p{x  -\-  x-^),  and  {y  —  ^/i)-  = 
2p{x  —  ^i). 


69,  Genebal  Scholium. — It  will  appear 
hereafter  that  the  conic  sections  are  formed  by 
the  mutual  intersection  of  a  plane  and  a  right 
cone  with  a  circular  base.  It  is  from  this  fact 
that  the  name  of  these  curves  is  derived ;  and 
from  this  as  a  definition  they  we^e  formerly 
studied.  The  different  species  arise  from  dif- 
ferent positions  of  the  cutting  plane.  The 
plane  which  gives  the  parabola  lies  parallel  to 
one  of  the  elements  of  the  cone,  as  O  N  P,  and 
hence  cuts  but  one  nappe,  and  gives  but  one 
branch.  To  produce  the  ellipse  the  cutting 
plane  lies  between  this  position  and  perpendic- 
ular to  the  axis,  as  fi~rmn.  To  produce  the 
hyperbola  it  lies  between  parallel  to  an  element 

and  parallel  to  the  axis,  as    HIK    and  EGF,  and  hence  cuts  both  nappes, 
giving  two  branches. 

Several  of  the  varieties  of  the  conic  sections  as  heretofore  considered,  may 
be  illustrated  by  means  of  this  geometrical  conception,  and  their  mutual 
relations  more  clearly  seen.  Thus  as  the  plane  of  the  ellipse  approaches 
perpendicularity  to  the  axis,  the  eHipse  approaches  the  form  of  a  circle  iijto 
which  it  passes  when  the  plane  becomes  perpendicular  to  the  axis.  The 
circle  is  therefore  a  variety  (or  more  properly  a  limit)  of  the  ellipse.  So 
also  as  the  plane  approaches  the  vertex,  the  ellipse  diminishes,  passing  into 
its  limit — a  point — when  the  x^lane  passes  through  the  vertex.  The  hyper- 
bola becomes  two  intersecting  straight  lines  when  the  cutting  plane  passes 
through  the  vertex  afid  is  not  parallel  to  an  element.  When  it  becomes 
parallel  to  an  element  and  also  passes  through  the  vertex,  it  gives  the  limit 
both  of  the  parabola  and  the  hyperbola,  which  common  limit  is  a  right 
line.  When  the  cone  passes  into  a  cylinder  the  parabola  becomes  two  par- 
a.lel  right  lines,  as  also  the  hyperbola  may,  if  it  is  conopived  as  produced  by 
a  cutting  plane  perpendicular  to  the  base.     If  the  cutting  plane  prod    cing 


52  THE   CARTESIAN  METHOD   OF   CO-OEDINATES. 

the  hypeibola  is  conceived  as  oblique  to  the  axis,  the  hyperbola  passes  into 
an  elhpse  when  the  cone  passes  into  a  cylinder. 

[Note. — Of  course  the  above  views  are  not  give  i  as  in  any  sense  needed  to  ccmfirm  the  conclu- 
sions of  the  preceding  discussions,  but  simply  to  give  the  student  a  Little  further  insight  into  the 
wonderful  harmony  which  exists  between  algebraic /ormMte  and  geometrical  loci.] 


or  five  arbitrary  condi- 


70,  IPTOp,  Through  fi\:)e  points  in  a  plane  one  conic  section  may 
always  he  made  to  pass,  and  but  one. 

Dem. — Dividing  the  general  equation  Ay"-{-  Bxy-^Cx-  -^Dy^Ejc-\-  F=  0  through 
by  F,  and  distinguishing  the  new  coefficients  by  accents,  we  have  A'y'^  4"  -S'a;?/ 
-{-C'x^--\-D'y  +  E'x-\-l=0.  Now  let  (x,,  y,),  {x.^,  y.z),  {x.^,  y.^),  {x^,  y^\  and 
(a; 5,  2/o)  be  the  five  given  points.  Substituting,  successively,  in  the  last  equation, 
the  co-ordinates  of  these  five  points,  for  the  general  co-ordinates  x  and  y,  there 
result  the  five  equations 

A'y,^-{-B'x,y,  -^  Cx^- -{- D'y,  -\-E'x,+l=X) 

A'y^^  +  B'x,y.z  4-  O'x,^  +  Dy,  -\-i:'x,+l=0 

A'y,^  +  B-x,y,  +  C'x,^  -f  D'y,  ^Kx,-^1=0 

A'y,2  +  B'x.y,  +  Cx,^  -f  D'y,  -f  Ex,  +  1=0 

A'y,2  -f  B'x.y,  +  Cx,^  +  Dy,  -{-Ex, +1=0 
tions.  This  number  of  conditions  is  possible,  since  there  are  Jive  arbitrary  constants 
involved  ;  viz..  A',  B',  C,  D,  and  E' .  From  these  equations,  asiC],  3/,,  iCg,  t/g* 
^■ii  2/3J  ^45  y^■>  ^.TJ  ^^d  y,,  are  known  quantities,  the  values  of  A' ,  B ,  C,  B',  and  E 
can  be  determined.  Having  found  the  values  of  these  coefficients,  by  substituting 
their  values  in  the  general  equation  A'y^  -\-  B'xy  -}-  C'x"  -)-  D'y  -\-Ex  -|-  1  =  0, 
there  results  an  equation  of  the  second  degree  between  two  variables,  or  an  equa- 
tion of  a  conic  section.  As  this  equation  is  satisfied  by  the  co-ordinates  of  each  of 
the  five  given  points,  the  locus  represented  by  them  passes  through  these  points. 
Finally,  as  the  five  equations  are  all  of  the  first  degree  with  respect  to  A',  B' ,  C, 
D' ,  and  E ,  but  one  set  of  values  can  be  determined  for  these  coefficients.  There- 
fore, hut  one  conic  section  can  be  made  to  pass  through  the  five  given  points,  q.  e.  d. 

ScH.  1. — In  this  proposition  the  term  Conic  Section  must  be  taken  in  its 
broadest  sense,  i.  e. ,  as  embracing  all  varieties  of  these  loci,  except  the  so- 
called  imaginary  loci. 

ScH.  2.  — If  the  five  points  are  so  situated  that  the  equation  of  the  locus 
passing  through  them  lacks  some  of  the  terms  of  a  complete  equation,  it  will 
not  do  to  divide  the  general  equation  by  the  coefficient  of  such  a  term.  If 
such  an  error  has  been  made  in  the  hypothesis  in  any  solution,  it  will  soon 
appear  as  the  solution  proceeds.  This  case  is  analogous  to  the  one  noticed 
in  the  suggestion  under  Ex.  4,  page  49. 

Ex.  1.  Produce  the  equation  of  a  conic  section  passing  through  the 
five  points  (2,  3),  (0,  4),  (—1,  5),  (  —2,  —1),  and  (1,  —2),  and  deter- 
mine its  species. 

Solution. — The  five  equations  which  determine  the  coefficients  A' ,  B',  (7,  D', 

and  E ,  are 


OP  THE   CONIC   SECTIONS.  53 

(1)  9 A'  -f  6B-  4-  46"  -f  3i>'  +  2^'  -f-  1  =0  ; 

(2)  16^'4-4i)'  4-1=0  ; 

(3)  25A'  —  55'  4-    C  -\-5lJ'  —    ^  +  1  =  0; 

(4)  ^'4-25' -I- 4C^  —   J>' —2^4-1=0; 

(5)  4.A' —2B'  +    C —2D' +    ^4-1=0. 

„  /.       ,,  ,.  .    ,  ,,  169     ^,  220     ^,         89      _       445 

Solving  these  equations,  we  find  ^  =  —  -^—,  B  =  —:^,  G  =  -— ,  D  =  — ^, 

113 

J?' =  — -.     Substituting  these  values  in  the  general  equation  J.'t/2 -}- 5'ic?/ 4"  C'a;^ 

4- -O'y- 4" -^'^  4~l=^j  clearing  of  fractions  and  changing  signs,  we  have  169?/- 
4-  220xT/  — •  89a;2  —  445?/  —  113x  —  924  ==  0,  which  is  the  equation  of  a  conic  section 
passing  through  the  five  given  points.  This  locus  is  an  hyperbola,  since  B-  —  4c.AC 
>0. 

Ex.  2.  Produce  the  equation  of  a  conic  section  passing  through, 
the  five  points  (1,  3),  (4,  —6),  (0,  0),  (9,  —9),  and  (16, 12),  and  find 
its  species. 

Suggestions. — As  one  of  the  given  points  is  (0,  0),  the  locus  passes  through  the 
origin  ;  and  hence  F=0.  The  form  of  the  general  equation  used  would,  there- 
fore, be  Ay^  -\-  Bxy  4-  Ox,''-  -{-  By  -j-  Ex  =  0,  which  divided  through  by  one  of  the 
coefficients,  as  A,  gives  the  form  y- -\-B'x^ -^C"x--\-B'y -]- U'x  =  0.  This  equa- 
tion satisfied  for  the  four  points  (1,  3),  (4,  — 6),  (9,  — 9),  and  (16,  12),  in  succession, 
gives  rise  to  four  equations  from  which  the  coefficients  can  be  determined. 

The  locus  is  a  parabola  whose  equation  is  j^  =  9x. 

Ex.  3.  Produce  the  equation  of  a  conic  section  passing  through 

(—4,  —2),  (2,  1),  (—6,  8),  (0,  0),  and  (2,  —1),  and  determine  its 
species.  The  equation  is  y  =  rp  ^x. 

Ex.  4.  Produce  the  equation  of  a  conic  section  passing  through 

(3,  \/5),  (—2,  0),  (—4,  — v/i2),  (3,  —Vl),  and  (2,  0),  and  deter- 
mine its  species.  The  locus  is  an  equilateral  hyperbola. 

Ex.  5.  Produce  the  equation  of  a  conic  section  passing  through 

i—h  —  i).  (2,  1),  (f,  2),  (— f,  —3)',  and  (f,  —  f)  and  determine 
its  species. 

The  locus  is  an  ellipse  whose  equation  is  y~  — ■  2xy  4-  3a72  -\-  2y  —  Ax 
—  3  =  0. 

Ex.  6.  What  is  the  equation  of  a  circle  whose  radius  is  5,  referred 
to  rectangular  axes,  and  the  origin  at  the  centre  ?  When  the  origin 
is  on  the  circumference  and  the  axis  of  abscissas  is  a  diameter? 
When  the  axes  are  tangent  to  the  circumference  ? 

Fquatio7is,  y^-{-x-^  =  25,  ?/2  =  db  10j7  —  x^,  and  y'^-^-x^  —  lOy  — •  10.77 
-f  25  =  0. 

Ex.  7.  What  is  the  equation  of  an  ellipse  whose  axes  are  16  and 
10,  when  referred  to  its  own  axes  ?     When  referred  to  its  transverse 


54 


THE  CARTESIAN   METHOD  OF  CO-ORDINATES. 


axis  and  a  tangent  at  the  left  hand  vertex  ?     The  corresponding  prob- 
lems in  the  case  of  the  hyperbola. 

Equations,  64i/2  _j_  25^2  =  1600.     64^/2  —  400^  +  25^72  =  0. 
64?/2  —  25^72  =  —  1600.     64i/2  +  400^  —  25^^  ==  0. 

SuQS. — The  results  of  tlae  two  preceding  examples  are  readily  written  from  the 
equations  of  the  respective  loci  as  given  in  (5S — 5T),  and  are  designed  to  fami- 
liarize those  most  important  forms. 

Ex.  8.  Produce  the  equation  of  a  parabola  referred  to  rectangular 
axes,  the  vertex  of  the  parabola  being  at  ( — 3,  — 2),  the  parameter,  6, 
and  the  axis  of  abscissas  parallel  to  the  axis  of  the  curve. 

Equation,  y^  +  4?/  —  Qx  — 14  =  0. 

Ex.  9.  Produce  the  equation  of  an  ellipse  whose  eccentricity  is  f, 
its  major  axis  18,  the  centre  being  at  ( — 2,  3),  and  the  axes  of  refer- 
ence being  rectangular  and  parallel  to  the  axes  of  the  curve. 

Equation,  9v-  +  '6x-'  —  54?/  +  20a7  —  304  =  0. 

Ex.  10.  What  are  the  following  loci,  and  what  their  axes  :  viz., 
9^2_|.4;j;2=,36?     7^^  +  ll?/2  =  15  ?     100y2_  25^2  =:_  2,500?    11  x-^ 

—  252/2=:=— 116? 


EXERCISES    IN    PRODUCING    THE   EQUATIONS   OF  THE   CONIC   SECTIONS 

FROM   OTHER   DEFINITIONS. 

[Note. — These  exercises  may  be  omitted  without  destroying  the  integrity  of  the  course.  They 
are  designed  simply  to  lead  the  student  to  a  more  full  comprehension  of  the  jprocess  of  producing 
an  equation  of  a  locus  from  its  definition,  a  subject  of  vital  importance  if  one  proposes  to  so  master 
this  method  of  geometrical  investigation  as  to  be  independent  in  the  use  of  it.] 

1.  To  produce  the  common  equation  of  the  ellipse  from  the  defini- 
tion : — The  ellipse  is  a  curve  such  that  the  sum  of  the  distances  from  any 
point  in  the  curve  to  two  fixed  points  called  the  foci,  is  constant  and  equal 
to  the  major  diameter. 

SuGS.       AD  =37,    PD=2/,    A&  =  A, 
AE.=B,  A  F  =  A  F'  =  c.    Then  from  the 


definition  \/y^  -\-ic  -\-  xy-  -\-  Vy'^  -{-[c  —  xY 
=  2^.  Whence  A"y-^  +  (^2  _  c2>2  =  j^2 
(^2  _  c;).  But  by  definition,  E.F  =  A  , 
whence  A^  —  c^  =  JB-  ;    and  we  have  A^^ 

2.  In  a  manner  similar  to  the  above  V 

produce   the    common    equation    of  Fig.  54. 

the  hyperbola,  from  the  definition,—  The  hyperbola  is  a  curve  such 
that  the  difference  of  the  didancesfrom  any  point  in  the  curve  to  two  fixed 
points  is  constant  and  equal  to  the  transverse  axis. 


OF   THE   CONIC!    SECTIONS. 


55 


Sxjg's. — In  this  case  it  must  be  borne  in  mind  that  A^-\-B'^=c^  {.4:7),  and  hence 
that  A^  —  c^  =  — B\     The  equation  is  A^y^  —  B^x^  =  — A^B^,  as  before  produced. 


3.  To  produce  the  equation  of  the  locus  of  a  point  moving  so  that 
the  square  of  its  distance  from  a  fixed  point  is  in  a  constant  ratio  to 
its  distance  from  a  fixed  hne. 


Y 

P 

X*  A  D  X 


Sug's. — Let  the  fixed  line  be  taken  as  the 
axis  of  abscissas,  and  let  a  perpendicular  to 
it  through  the  fixed  point,  F,  be  taken  as  the 
axis  of  ordinates.  Let  P  be  any  point  in  the 
locus.  Then  AD=.r,  and  PD=?/.  As 
A  F  is  constant,  call  it  a,  and  let  m  repre- 
sent the  ratio  referred  to  in  the  definition  in  the 
example.     The  equation  sought  is  {y  —  a)'^  -j"  *''  Iv 

=  my,  or  y-  -f-  x~  —  (2a  -f-  'm)y  -f-  a-  =  0.     This  I''ig.   55. 

being  an  equation  of  the  second  degree,  the  locus  is  a  conic  section.  Again,  as 
-B2  —  4J.C<<  0,  it  is  an  ellipse.  Finally,  as  the  coefficients  of  y'^  and  x^  are  equal, 
it  is  an  ellipse  with  equal  axes,  or  a  circle. 

To  determine  more   fully  the   situation  of  this    circle,  notice  that   for  y  =  0, 
flj  =  =b  \/—  a'~,  whence  we  see  that,  in  general,  the  circle  does  not  cut  the  axis  of  x. 
2(7,  -f-  tn 


Making  x  =  0,  ?/ 


iam  -f-  w^ 


Now,   as   every  value  of  y  in   the 


■A  \^  4 

equation  of  this  locus  gives   tv»^o  values  of  x,  numerically  equal  but  with  oppo- 
site signs,  we  see  that  the  locus  is  symmetrical  with  the  axis  of  y,  and  that  the  cen- 

tre  of  the  circle   lies   in   this   axis.     But   the   circle   cuts   this   axis   at    ' 


>^ 


,'  t(a)i  4-  m'        -     ,   2rt  4-  m 
-f-      I 4 and  at 


, : .     Whence   the    diameter   is  the 

4  2  \l  4: 

difi"erence    between    these    values  ;     and    letting    r    be    the     radius,    we     have 
1    .- 


r  =--^.1.6(7/1  -\-  'III ,  i.  e.,  the  radical  part  of  the  root. 

In  the  particular  case  in  which  a  =0  ;  i.  e.,  when  F  is  at  A,  the  equation  be- 
comes y^  -f-  X'  :=  my,  which  is  the  equation  of  a  circle  passing  through  the  origin, 
and  having  its  centre  on  the  axis  of  y. 


4.  In  the  given  right  Hnes  A  P,  A  Q,  intersecting  at  right  angles, 
are  taken  variable  points  p,  q,  such  that  Ap  I  7?  P  '.'.  QiQ  '.  qA  ; 
prove  that  the  locus  of  the  intersection  of  P^,  Qj),  is  an  ellipse  which 
touches  the  right  lines  in  P  and  Q. 


56 


THE   CARTESIAN   METHOD   OF   CO-ORDINATES. 


Sug's.— Let  AP  and  AQ  be  taken  as  the  axes 
of  reference.  Call  A  P  =  a,  and  AQ  =  6.  Then 
AD  =x,  and  RD  =2/-  ^rom  the  similar  trian- 
gles RPD,  5 PA,  and  RpD,  QpA  obtain  the 
relation  between  x,  y,  a,  and  5,  This  wiU  be  the 
equation  sought.  It  is,  when  reduced,  a\y—hY 
j^  l,i(^x  —  aY  =  «"^?>'^  —  <^^^y- 

.ScH. — As  this  equation  is  of  the 
second  degree,  and  B^ — 4:AG^ 
—  3a2&2,  the  locus  is  an  ellipse. 
As  there  is  a  term  in  xy,  the  axis 
of  the  curve  is  inclined  to  the 
axis  of  abscissas.  For  x  =  a,  p 
y  ^=h  and  0 ;  hence  the  locus 
passes  through  [a,  h),  and  P. 
I'or  a;  —  0,  3/  =  6  ;  therefore  the 
locus  passes  through  Q. 

To  effect  the  construction  mechanically,  take  h.p  '.  pP  :  :  Qg  :  gA,  by- 
composition  and  alternation,  giving  AP  :  AQ  :  :  Pi?  :  Ag.  Now  assuming 
p  at  any  point  in  AX,  we  can  find  the  corresponding  value  of  Ag.  After 
p  passes  P,  Ag  becomes  —,  and  is  laid  off  below  A.  So  when  Q_p'  passes 
parallelism  with  AX,  Ap  becomes  negative. 

5.  Required  the  locus  of  the  middle  point  of  a  line  moving  with  its 
extremities  in  two  fixed  lines  at  right  angles  with  each  other,  while 
it  passes  through  a  fixed  point. 

Sug's.  —Take  the  fixed  fines  as  axes  of  refer- 
ence. Let  O  ]3e  the  fixed  point,  and  C  B  the 
line,  the  locus  of  whose  centre,  P,  is  to  be  de- 
termined. Calling  AD  «,  and  OD  &,  the 
equation  of  the  locus  is  'i.o^y  —  ay —  'bx=Q, 
which  is  the  equation  of  an  hyperbola,  passing 
through  the  origin,  since  for  .^  =  0,  2/  =  ^;  and 
also  passing  through  O,  since  x  =  a,  gives  y 
=  &.     Let  the  piipil  trace  the  curve. 


Fig.  57. 


6.  Required  the  locus  of  the  point   P, 
moving  so  that  P  D  ,    bears    a   constant 
ratio   to    A  D  X  D  B  ;    A  and    B  being 
fixed   points.     What  is 
this  ratio  is  1  ? 


and 
the  locus  when 


The  distance    A  B   heing  called  2a,  and  ^i»-  ^^^ 

the  ratio  m,  the  equation  is  y2  =  mx(2a  —  x),  whence  the  locus  is  seen 
to  he  an  ellipse.     If  m=l,  it  is  a  circle. 


EQUATIONS   OF   HIGHER   PLANE   CUEVES.  57 


7.  If  P  moves  in  Fig.  58,  so  that   PD^  bears  a  constant  ratio  to 
A  D,  what  is  the  locus  ? 


-4-»-^ 


SUCTION  VIZ 
Equations  of  Higher  Plane  Curves. 

71,  One  variable  is  called  a  Function  of  another  variable  when 
it  depends  upon  that  other  variable  for  its  value.  Thus  the  ordinate 
of  a  curve  is  a  function  of  the  abscissa. 

72,  Functions  are  classified  as  Algebraic  and  Transcen- 
defltalf  and  the  latter  are  subdivided  into  TrigonOTnetvic. 
and  Circular,  Logarithmic,  and  Eocponential, 

73,  An  Algebraic  Function  is  one  which  involves  only  the 
elementary  methods  of  combination,  viz.,  addition,  subtraction,  mul- 
tiplication, division,  involution  and  evolution.  Thus  in  ?/  =  ax^  —  3^^ 
and  in  all  the  equations  hitherto  discussed  in  this  chapter,  y  is  an 
algebraic  function  of  x,  except  24-33,  Sec.  II. 

74=.  A  TrigonometJ'ical  Function  is  one  which  involves 
sines,  cosines,  tangents,  cotangents,  etc.,  as  2/  =  sin  x,  y=  sin  x  tan 
X,  etc. 

75,  A  Circular  Function  is  one  in  which  the  concept  is  an 
arc  (in  the  trigonometrical  the  concept  is  a  right  line).  These  are 
written  thus  :  2/==sin~^^,  read  "?/  equals  the  arc  whose  sine  is  .a?"; 
2/=tan~^j7,  read  "?/  equals  the  arc  whose  tangent  is  ^." 

Notice  that  in  the  expression  2/  =  tan~'a:,  it  is  the  arc  which  we  are 
to  think  of,  while  in  the  expression  ^=tan?/  it  is  the  tangent,  which 
is  a  right  line.  Trigonometrical  functions  are  right  lines  ;  circular  ^ 
functions  are  arcs.  These  functions  are  mutually  convertible  into  \ 
each  other  ;  thus,  y  =  s,m.~'^x,  is,  equivalent  to  ^  r=  sin  ;?/,  the  only 
difference  being  that  in  the  former  we  think  of  the  arc,  the  sine  being 
given  to  tell  what  arc,  and  in  the  latter,  we  think  of  its  sine,  the  arc 
being  given  to  tell  what  sine. 

The  circular  functions  y  =  ^mr^x,  y  =  cos~^^,  y  =  seG~^x,  etc.,  are 
often  called  the  Inverse  Trigonometrical  Functions. 

76,  A  LogaritJiTnic  Function  is  one  which  involves  loga- 
rithms ;  as  2/  =  log  x,  log^  i/  =  3  log  ax,  etc. 

77,  An  Fxponential  Function  is  one  in  which  the  varia- 
able  occurs  as  an  exponent ;  as  i/  =  a"",  z  =  x^,  etc. 


58 


THE   CAHTESIAN   METHOD    OF   CO-ORDINATES. 


78.  Higher  Plane  Curves  are  loci  whose  equations  are 
above  the  second  degree,  or  which  involve  transcendental  functions. 
As  it  has  already  been  shown  that  loci  of  the  equations  of  the  1st 
degree  are  right  lines,  and  that  loci  of  the  2nd  degree  are  conic  sec- 
tions, it  follows  that  all  other  j^lane  loci  are  higher  plane  carves. 
The  former  are  called  Lower  Plane  Loci. 

Of  course  the  variety  of  higher  plane  loci  is  infinite.  We  can  con- 
sider but  a  few,  and  these  simply  as  specimens. 

79.  An  Algebraic  Curve  is  one  whose  equation  contains  only  alge- 
braic functions.  A  Transcendental  Curve  is  one  whose  equation  con- 
tains transcendental  functions  ;  when  converted  into  algebraic  forms 
their  degree  is  infinite. 


THE   CISSOID   OF  BIOCLES. 

50.  Def.  If  pairs  of  equal  ordinates  be  drawn  to  the  diameter 
of  a  circle,  and  through  one  extremity  of  this  diameter  and  the  point 
in  the  circumference  through  which  one  of  the  ordinates  is  let  fall,  a 
line  be  drawn,  the  locus  of  the  intersection  of  this  line  and  the  equal 
ordinate,  or  that  ordinate  produced,  is  the  Cissolcl  of  U lodes. 

51.  IPvoh.     To  construct  the  Cissoid. 

Solution. — Let  AB  be  the  diameter  of  a  circle  ;  and  ED,  and 
ED'  be  equal  ordinates.     Through  A  and  E'  draw  AE'  inter- 
secting ED  in  p.     Then  is  P  a  point  in  the  locus.     In  hke 
manner  draw  AE  and  produce  it  till  it  meets  E'D' 
produced  in  |.     Then  is  |  a  point  in  the  locus.     In 
the  same  way  other  points  are  found  both  above  and 
below  AB.     There  are,  therefore,  two  branches  of  the 
locus  ACM  and  ACM',  symmetrical  with  respect 
to    the    diameter    AB-      These    branches    evidently 
meet   at   A,  pass   through  the  extremities  of  the  di- 
ameter  CC,   and  have   GG'  as  a  common  asymp- 
tote. 

ScH.  1. — The  name  Cissoid  is  from  the  Greek 
and  signifies  ivy-form.  It  was  applied  to  the 
curve,  probably,  from  its  resemblance  to  the 
graceful  outline  presented  by  a  growth  of  ivy 
upon  a  wall.  The  locus  was  invented  by  the  Greek 
geometer  whose  name  it  bears,  while  he  was  seeking 
the  solution  of  the  celebrated  problem  of  the  Daplication 
of  the  Cube. 

Sen.  2. — Sir  Isaac  Newton  gave  tho  following  mechan- 
i'^a'    m>rr)l    of    dMScrihinc^  this    locus:    Lot    AI3    l)e  the 


EQUATIONS  OF  HIGHER  PLANE  CURVES. 


59 


diameter  of  the  circle  from  which  the  curve  would  be 
described  by  the  definition  ;  at  the  centre  O  erect  the 
perpendicular  OL.  and  take  AD  =  AO  =  OB.  Now 
take  a  rectangular  ruler  FEC,  whose  leg  CE  =  AB,  and 
Avhile  the  extremity  C  moves  in  the  line  OL.  let  the  leg 
FE  sUde  through  the  fixed  point  D,  then  will  the  middle 
point  of  CE»  P>  describe  the  cissoid.  [The 
demonstration  will  afford  a  good  exercise  for  the 
student.  ] 

ScH.  3.  — This  curve  is  also  the  locus  of  the 
vertex  of  a  common  parabola  rolling  upon  an 
equal  parabola. 


LI 


Fig.  60. 


82,  JProh, — To  produce  the  equation  of  the  Cissoid  of  Diodes. 

Solution. — In  Fig.  59  let  AX  and  AY  be  the  axes  of  reference,  AB  =  2a, 
the  diameter  of  the  circle  referred  to  in  the  definition,  and  P  any  point  in  the 
curve.  Then  AD  =  x,  and  PD  :=?/.  Draw  through  P  the  ordinate  ED,  and 
also  draw  the  equal  ordinate  E'D'.     APE'  is  a  straight  Une  by  definition.     We 

Squar- 


now  have   AD  :  P D  :  :  A D ' 
ing  and  reducing,  x-  :  y-  : :  2a 


E'D',  or  ic  :y  :  :2a  —  a*.  :  \/(2a 


X  :  X. 


yi 


X^ 


'Aa  —  X 


is  the  equation  sought. 


ScH.  1. — Since  y  = 


-, every  real  +  value  o/"  x  <<  2a  gives  two  real 


'2a  —  x' 

and  numerically  equal  values  of  y,  loiih  contrary  signs.  Hence  the  locus  is 
symmetrical  with  respect  to  the  axis  of  x.  For  x  =  2a,  y  =  ±:  go  ,  whence  the 
brandies  are  infinite,  and  GG'  is  an  asymptote  to  both  branches.  For  all 
values  ()fx'^  2a,  and  for  :l  negative,  j  is  imaginary.  Therefore  the  locus  is 
comprised  between  the  limits  x  =  0,  x  =  2a. 

ScH.  2. — By  the  Duplication  of  the  Cube  is  meant  finding  the  edge  of  a  cube 
which  shall  have  twice  the  volume  of  a  cube  whose  edge  is  given.  To  efiTect  this 
by  means  of  this  curve,  lot  AM  bo  any  cissoid,  AB  the 
diam(!ter  of  the  circle  which  pertains  to  it,  and  O  the  centre 
of  that  cii'cle.  Take  C0=20B,  and  draw  CB.  Let 
fall  from  the  jDoint  P,  where  C  B  cuts  the  curve,  the  per- 
pendicular P  K.  Then  P  K  =  2  B  K.  Now  a  cube  des- 
cribed on  P  K  is  twice  one  described  on  A  K  ;  for 
since  P  K  =  y,  A  K  =  a;,  and  K  B  =  2a  —  x,  we  have 


ak' 


AK^ 


AK*. 


PK 


PK"  =  -  —  =  1: or   iPK 

KB        IPK' 

=  2AK  .     Finally,  let  a  be  the  edge  of  any  given 

cube  ;  fmd  r/,  so  that  a:ai  ::  AK  :  PK,  whence  a^  :  c/i^  :  AK^ :  PK\      But 


A         OK 
Fig.   01. 


PK  =2AK. 


ai3  =  2a3. 


60 


THE   CAETESIAN    METHOD    OF    CO-OKDINATES. 


Bj  taking  CO  =30B  aud  proceeding  in  a  similar  manner,  we  can  tri- 
plicate the  cube;  or  in  the  same  way  obtain  the  edge  of  a  cube  of  any  given 
number  of  times  the  volume  of  a  given  cube.  (The  pupil  may  show  that 
|"k'  =: 2Kb' ;  also  that  Ak'  =  ^Tk^) 


THE  CONCHOID  OF  MCOMEDES. 

83.  r>EF. — The  Conchoid  of  Wicojuedes  is  the  locus  of  a 
point  in  a  line  which  revolves  on  and  slides  in  a  fixed  pivot,  so  as  to 
allow  a  constant  portion  of  the  line  to  project  beyond  a  fixed  right  line. 


84,  JProh,     To  construct  the  Conchoid  of  Mcomedes. 


Solution.  —Let  O*  be  the 
fixed  point,  or  pivot,  X'X 
the  fixed  Hne,  and  A  B  the 
constant  portion  of  the  re- 
volving line.  Draw  a  con- 
venient number  of  radiating 
lines  through  O,  and  on  each 
lay  ofl"  above  X'X  the  dis- 
tances CI,  FP,  E6,  etc., 
equal  to  A  B.  Then  will  1, 
2,  3,  4,  etc  ,  be  points  in  the 
locus  ;  and  M  B  N  will  be  the  conchoid. 

ScH. — This  locus  is  readily  drawn  by  mechanical  means.  Let  X'X  and 
YY'  be  two  bars  fixed  at  right  angles  to  each  other.  Let  any  one  of  the 
radiant  lines,  as  OP,  represent  a  ruler,  grooved  on  the  under  side  so  as  to 
slide  on  the  head  of  a  pin  fixed  in  the  bar  YY',  at  O.  Let  there  be  a 
fixed  pin  on  the  under  side  of  the  ruler,  as  at  F,  which  can  slide  in  a 
groove  on  the  upper  side  of  the  bar  X'X.  Now,  placing  the  groove  in  the 
rnler  on  the  head  of  the  pin  at  O,  and  the  pin  in  the  ruler,  in  the  groove  in 
XX,  any  point  in  the  ruler,  as  P,  will  describe  the  conchoid. 


8S.  JPvoh.     To  produce  the  equation  of  the  Conchoid  of  Nicomedes. 

Solution.  —Let  P,  Fig.  62,  be  any  point  in  the  locus  referred  to  the  axes 
XX',  Y  Y';  and  let  its  co-ordinates  A  E  and  PE,  be  tc  and  y.  Let  A  B  =  a,  and 
AO  =  6.  Produce  PE  till  it  meets  OD  drawn  parallel  to  AX.  Now,  by  simi- 
lar triangles,  PE  :PD  ::EF  :OD;  or  y  -.y-}-})  ::  Va^  —  2/2  :  x.  Squarmg, 
2/2  :  (2/  -f-  &)2  : :  a-2  _  2/2  :  a;2.     ...  xHf-  =  {y  -\-  hy{a^  —  y^). 

ScH.  1. — Since  .-r  =  ±  v/a^  —  yy- j,  for  every  positive  value  of  y, 

numerically  less  than  a,  x  has  two  nnmerically  equal  values  with  opposite 
signs  ;    which  values  increase  as  y  di7ninishcs,   and  for  ?/  ==  0.  .r  =:  dz  ex. 


EQUATIONS  OP  HIGHER  PLANE  CURVES. 


61 


.  • .  This  portion  of  the  locus  is  symmetrical  with  respect  to  the  axis  of  y, 
and  has  the  axis  of  x  for  a  common  asymptote  of  its  two  branches.  Again, 
as  ail  negative  values  of  y,  not  numerically  greater  than  a,  give  numerically 
equal  values  of  x  with  opposite  signs,  there  is  a  portion  of  the  locus  below 
the  axis  of  .r,  wliich  is  also  symmetrical  with  respect  to  the  axis  of  y.  To 
discover  the  form  of  this  portion,  1st  consider  a  >  h.  Then  f or  y  =  —  a, 
or  — 5,  a?  =  0,  but  for  values  of  y  between  these  two  limits,  x  has  two  nu-; 
merically  equal  values  with  opposite  signs  ;  hence  the  locus  between  these 
two  limits  is  an  oval  symmetrical  with  respect  to  the  axis  of  y.  For  y  nu- 
merically less  than  h,  and  negative,  the  values  of  x  increase  numerically  till, 
at  3/  =:  0,  they  become  =h  oo  ;  hence  between  O  and  the  axis  of  abscissas 
there  arc  two  infinite  branches,  symmetrical  with  respect  to  the  axis  of  y, 
and  having  the  axis  of  x  as  a  common  asymptote.  2nd.  When  a^=b  the 
oval  disappears.  These  forms  are 
described  mechanically  by  taking  the 
point  on  the  moving  ruler  below  the 
fixed  line. 


ScH.  2. — When  h  =  Q,  the  equation 
becomes  .r-y^  ^^  y'^{a^  —  y'^) ;  or  ic^  _|,  y2 
=  a'^.  This  is  the  equation  of  the 
circle,  as  it  evidently  should  be. 

ScH.  3. — This  curve  was  invented 
by  the  geometer  whose  name  it  bears, 
for  a  purpose  similar  to  that  subserved 
by  the  cissoid.  The  problem  of  the 
Duplication  of  the  cube  and  the  Tri- 
section  of  an  angle  had  been  shown  to 
be  identical,  as  both  depend  upon  the 
insertion  of  two  means  in  a  continued 
proportion  between  two  extremes. 
Thus,  letting  a  and  b  be  the  extremes, 
it  is  required  to  find  x  and  y,  so  that 
a  :  X  :  y  :  b  ;  i.  e.,  a  :  x  :  :  x  :  y,  and 
X  :  y  : :  y  :  b.  This  problem,  viz.,  the 
insertion  of  two  means  between  two 
extremes,  is  effected  by  the  cissoid. 
In  the  cissoid,  I^ig.  59,  ED  and  AD' are 
the  two  means  between  AD  and     ID'  : 


Fig.  63. 


Fig.  65. 


that  is,  AD   :  ED   :  AD'  :  ID'. 

The  Triseciion  of  an  angle  by  means  of  the  conchoid  is  effected  thus  :  Let 
COM,  Fig.  66,  be  the  angle  to  be  trisected.  From  any  point,  D,  in  one 
leg  let  fall  a  perpendicular,  DB,  on  the  other.  Take  CB  =2D0,  and 
with  O  as  the  fixed  point,  X'X  as  the  fixed  line,  and  CO  as  the  ruler  with 
the  constant  portion  CB  projecting  beyond  X'X,  construct  the  arc  CR  of 
the  conchoid.  Erect  DP  perpendicular  to  X'X,  and  draw  PO.  Then  is 
POC  one-third  of  COM.     To  prove  this,  bisect  PH  as  at  E,  and  draw 


62 


THE   CARTESIAN   METHOD   OF   CO-ORDINATES. 


DE.  Draw  also  FE  parallel  to  DH.  Since 
PE  =  EH,  PF=  FD,  and  ED  =  PE  =  EH  = 
DO-  By  reason  of  the  isosceles  triangles  RED, 
and  EDO,  we  have  angle  DEO  =  2P  =2POC. 
But  DEO  =  EOD.  .".  2EOC  =  EOD,  or 
EOC  =  iCOM. 

[Note  — This  scholium  is  by  no  means  necessary  to  tlie  in- 
tegrity of  the  course.  It  is  inserted  merely  as  a  matter  of 
interest  to  the  student,  giving  him  a  few  hints  upon  a  subject 
which  has  figured  so  prominently  in  the  history  of  geometry. 
It  vnll  afford  a  good  exercise  for  the  student  who  has  time 
and  abiUty,  to  demonstrate  fuUy  the  facts  hinted  at,  and  which 

are  not  demonstrated  above.  Thus,  let  him  show  why,  in  the  cissoid,  AD  :  ED  :  AD'  :  ID';  also 
how  the  insertion  of  two  means  enables  us  to  obtain  any  multiple  of  the  cube  ;  also  how  the 
conchoid  effects  the  same  pvirpose  ;  and  that  the  two  problems  are  in  reality  but  one.] 


F 

) 

c 

^^^^ 

\ 

^  ^ 

\ 

M\ 

F 

-\e 

x' 

C 

^ 

B 

0 

X 

Fig.  66. 


THE  WITCH  OF  AGKESI. 

S(y.  Def. — Hie  Witch  of  Agnesi  is  the  locus  of  the  extrem- 
ity of  an  ordinate  to  a  circle,  produced  until  the  produced  ordinate 
is  to  the  ordinate  itself,  as  the  diameter  of  a  circle  is  to  one  of  the 
segments  into  which  the  ordinate  divides  the  diameter, — these  seg- 
ments beinsf  all  taken  on  the  same  side. 


87*     I*TOh,     To  condruct  the  Witch  of  Agnesi. 


SoLTJTioK. — Let  AB  be  the  circle 
Draw  a  series  of  parallel  ordinates  1  1 
2  2,  P'P,  etc.  To  find  a 
point  in  the  locns,  take  P  E 
:  F  E  :  :  A  B  :  A  E,  and  P 
is  siicli  a  point.  In  hke 
manner  locate  other  points, 
a3 1,  2,  etc. 


88,     I*TOh,     To  produce  the  equation  of  the  Witch. 

Solution.— Letting  the  axes  be  as  represented  in  Fig.  67,  so  that  P  being  any 
point,  A  D  =  cc,  P  D  =  j/'  ^^^  calling  the  diameter  of  the  circle,  A  B  =  2a,  the 
equation  is  a>!?/  =  4a-(2a  —  y).     [Let  the  student  supply  the  demonstration.] 

ScH. — The  Witch  has  but  one  portion,  as  represented  in  the  figure  ;  it  is 
symmetrical  with  respect  to  the  axis  YY',  is  comprised  between  y  =0,  and 
y  =  2a,  and  has  X'X  for  an  asymptote.      [Let  the  student  give  the  proof.] 


EQUATIONS  OP  HIGHER  PLANE  CURVES. 


63 


THE  LEMNISCATE  OF  BERNOUILLI. 

89,  Def. — The  JLemniscate  of  JBernoiiilli  is  a  curve  such 
that  the  product  of  two  Hues  drawn  from  any  point  in  it  to  two  fixed 
points,  called  the  foci,  is  equal  to  the  square  of  half  the  distance  b^ 
tween  these  foci. 


do.  JProh.  To  construct  the  Lemniscale  of  BernouilU. 

Solution. — Let  F  andF'  be  the  foci.     From 
F'  as  a  centre,  with  any  convenient  radius,  as 
F'  P,  draw  an  arc,  as  P6,    Find  a  third  propor- 
tional to  F'Pand  F'A.  Let  this  be  PF.  From 
F  as   a   centre   with   this   third  proportional, 
draw  an  arc  intersecting  the  former  in  P  and 
6.     Then  will   P  and  6  be  points  in  the  locns  ;  for,  by 
construction    F'P  X   PF  =  AF-.     In  like   manner  find 
other  points.     Fig.  G9,  shows  a  convenient  method  of  find- 
ing these   proportionals.     GH  =  F'F,  and  XG  =  AF. 
Since    TL  :  TG  :  :  TG   :  Tl ,  XL  :  AF  :  :  AF  :  T 
and  T  L  and  T I   are   the  corresponding  radii   to  be  used 
in  locating  a  point,  as  P.     With  one  pair  of  distances  four 
points  can  be  found. 


01,  Pvoh.   To  produce  the  equation  of  the  Lemniscate. 

Solution.— Assuming  X'X'  and  YY'  as  axes  of  reference,  letting  P  be  any 
point  whose  co-ordinates  AD  and  PD,  are  x  and  ^,  and  putting  A F  =  A F' 
=  c,  we  have  the  distance  between  the  two  points  F'  and  P,  or  F'P 
=  \/{x-\-cr-\-y-.  In  like  manner  FP  =  \/{c  —  xY  -\-  y-.  Whence  by  definition 
\/i^~cjH^^  X  V{G—xY-\-y^  =  G%  or  (2/'-'  -f  x^f=  2c^(x'-^  — •  y^). 

ScH.  1. — Let  the  student  observe  the  symmetry  and  limits  of  the  curve 
from  its  equation.  Observe  that  in  the  construction  F'B  =:FC  =  TW- 
As  FB  X  F'B  =c2,  and  FB  =  AB  — c,  and  F'B  =  AB  +  c,  we  find  that  AB 
=Cv/2.  Putting  AB  =-a  =  c\/%  2c2  =  a?,  whence  the  equation  of  the  curve 
in  terms  of  its  semi-axis  is  (3/2  J^x'^Y'=  ^^{^^  — 3/-)- 

ScH.  2. — (To  be  read  on  review.)  The  equation  of  the  Equilateral  Hy- 
perbola whose  semi-axis  is  a,  and  co-ordinates  x',  y' ,  is  x''^ — y"^  ^a^.  The 
equation  of  its  tangent  is  xx  —  yy'  =  a^.     The  equation  of  a  perpendicular 

x' 
from  the  centre  upon  the  tangent  isx  = y.     From  these  three  equations, 

ehminating  x'  and  y',  that  is  finding  the  locus  of  the  intersection  of  a  per- 


64 


THE   CARTESIAN   METHOD   OF   CO-ORDINATES. 


pendicular  from  the  centre  upon  tlie  tangent,  we  find  [x-  -{-  3/2)2  =  a-(x^  —  y^). 
Therefore  this  lemniscate  is  the  locus  of  the  intersection  of  a  perpendicular 
from  the  centre  of  an  equilateral  hyperbola  upon  its  tangent,  the  axes  of 
both  loci  being  coincident. 


THE  CYCLOID. 

92.  Def. — TJie  Cycloid  is  the  locus  of  a  point  in  the  circum- 
ference of  a  circle  which  rolls  along  a'  fixed  right  line. 


ScH.  — The  cycloid  can  be  constructed 
mechanically  by  rolling  a  wheel,  as 
HPI,  Fig.  70,  along  the  edge  of  a 
fixed  ruler,  as  AX.  A  point  P  in  the 
circumference  of  the  wheel  describes 
the  cycloid. 


Fig.   70. 


03.  Def's. — The  circle  H  P I  is  called  the  Generating  Circle, 

or,  simply  the  G-eneratrix  /AX  is  the  J^ase,  and  is  equal  to  the 
circumference  of  the  generatrix ;  and  B  F,  erected  perpendicular  to 
the  base  at  its  centre,  is  the  A-QCis,  and  is  equal  to  the  diameter  of 
the  generatrix. 


04:.  JProh, — Having  given  the  cycloid,  to  put  the  generating  circle  in 
position. 

Solution. — There  is  given  simply  the  curve  A  BX,  Fig,  70.  Draw  the  base 
AX,  and  bisect  it  by  the  perpendicular  BF.  BF  is  the  axis.  Bisect  the  axis 
by  N  K  drawn  parallel  to  the  base.  Now,  to  put  the  generating  circle  in  the  posi- 
tion it  occupied  when  the  generating  point  was  at  P.  draw  from  P  as  a  centre, 
with  a  radius  equal  to  the  radius  of  the  generatrix  (BO  or  OF),  an  arc  cutting 
N  K,  as  at  C.     C  is  the  centre  of  the  generatrix. 


Oo.  I*roh. — To  produce  the  equation  of  the  cycloid  referred  to  itx 
base  and  a  perpendicular  at  the  left  hand  vertex. 

Solution. — Let  P,  Fig.  70,  be  any  point  in  the  cycloid  A  BX,  referred  to  A  V, 
and  AX  as  axes.  Then  AD  =cc,  and  PD  =2/.  Call  the  radius  of  the  generatrix 
r.    Now  A  D  =^  A I  —  D  I .     But  by  construction,  A I  =  arc  P I  =  versin  —  '  I  L, 


Dl  =  PL 


or    versin  ~^t/,    to    a   radius  r. 

z=  \/'2ry  —  y^.       • .  cc  =  versin—^  y  —  \/2ry  — 


2/) 


y2 


ScH. — If  y  be  negative,  \/2ri/  —  ?/-'  becomes  imaginary  ;  hence  the  curve 
lies  on  but  one  side  of  the  base.     For  y  =0,  we  have  x  r^r.  versin  -^0  =  0, 


EQUATIONS  OF  HIGHER  PLANE  CURVES. 


65 


2itr,  4iTtr,  etc. ,  etc.  Hence  we  see  that  there  are  an  infinite  number  of  arcs 
like  ABX,  belonging  to  the  curve.  This  is  also  apparent  from  the  defini- 
tion, as  each  revolution  of  the  generatrix  produces  one  arc,  and  there  is  no 
limit  to  the  number  of  revolutions.  For  y  =  2r,  a;  =  versin  ~^(2r)  =  ^r, 
S^rr,  bTtr,  etc. ,  etc. ,  as  it  should  from  the  construction. 


OS,  JProb. — To  produce  the  equation  of  the  cycloid  referred  to  its 
axis  as  the  axis  of  abscissas,  and  a  tangent  at  the  vertex  of  the  axis  (  B,  ^ig. 
71),  as  the  axis  of  ordinates. 

Solution.— Let    PM  =?/,    and   BM  =ic.     Now  PM=PL+   LM.     But 

PL-=\/2r.^• — x^,   and   LM  =  IF  =  AF  — Al=the    semi-circumference    of 
the   generatrix  —  arc  PI.       Again,    arc  PI    ==:  versin  — i  (2r  —  x).      .*.    y  = 

\/2rx  —  x"^  -\-  7tr  —  versia— 1  (2r  —  jc). 


97.  Cor.    PR  =  arc    BR, 
■ing  any  point  in  the  curve. 


P  he- 


For    PR  =  L  M  =  A  F  —  A I   =  arc 
HPI  — arc  PI  =arc  HP  =  arc  BR.  A  I  F 

Fig.  71. 

ScH.  1. — Considering   the   equation   x  =  versin-'y  —  \/%-y  ~~y^,    we 

observe  that  there  are  an  infinite  number  of  values  of  x  for  every  value  of 
y.  First  of  all,  the  term  versin-^  is  ambiguous  as  to  its  sign,  since  a  nega- 
tive arc   has  the   same  versed-sine  as  the  numerically  equal  positive  arc. 

Moreover,  whatever  a  versed-sine  may  be,  there  are  not  only  the  -j-  and 

arcs  less  than  180°,  and  also  the  +  and  —  arcs  of  360°  —  the  former,  which 
corresponds  to  it,  but  these  increased  numerically  by  every  multiple  of  lit. 
We  are  therefore  to  write  the  term  versin~'y  with  the  sign  d=:,  and  under- 
stand that  for  every  value  of  y  it  has  an  infinite  number  of  numerical  values, 
each  succeeding  value  in  the  series,  being  numerically  lit  greater  than  the 
preceding.     In  the  second  place  the  term  —  s/'lry  —  y^,  being  a  square 

V2ry  — y2. 

-  y2ry 


root  is  to  be  written  —  (d=:\/2?^y  —  y"^),  or 

tion  in  this  way  we  have  x  =^  ±i  versin~'y 
siefnificance  of   these  facts  is  as   follows. 


Writing  the  equa- 
y'.     The  geometrical 


Let  y  have  any  value  as  PD, 
Fig,  70  ;  then  x  has  1st,  the  positive  value  AD  and  gives  the  point  P,  and 
a  numerically  equal  negative  value,  giving  a  point  P'  similarly  situated  on 
the  left  of  AY,  if  we  take  versin"'  y  <  180° ;  but  if  we  take  360°  —  this 
arc,  both  -f  and  —  as  the  value  of  versin— 'y,  we  get  two  other  values  of  x, 
one  -f  and  the  other  — .  The  former  is  where  PM,  Fig.  70,  produced  to 
the  right  meets  the  curve,  and  the  other  the  corresponding  point  on  the 
left  of  AY.  Taking  for  values  of  versin— ly,  the  values  now  considered 
-\-  l7t  simply  repeats  the  curves  at  a  distance  27t  both  at  the  right  and  left. 
The  equation  in  (.96)  has  a  similar  interpretation. 


66 


THE  CARTESIAN   METHOD   OP  CO-ORDINAIES. 


ScH.  2.  — The  Cycloid  is  a  transcendental  curve,  and  is  next  to  the  Conic 
Sections  in  importance  among  plane  loci. 

OS,  It  frequently  occurs  that  the  equation  of  a  locus  can  be  writ- 
ten immediately  from  the  definition.  The  Sinusoid  is  of  this  char- 
acter. The  definition  is,  "  The  Sinusoid  is  the  locus  of  a  point  whose 
abscissa  is  the  arc,  while  its  ordinate  is  the  sine  of  the  arc."  Hence 
the  equation  is  y  =  sin  x.  The  other  simple  trigonometrical  curves 
(page  16,  Ex's  27 — 33)  are  of  the  same  character,  as  is  also  the  loga- 
rithmic curve,  X  =  log  y. 

00,  As  has  been  remarked  before,  the  number  of  plane  curves  is 
infinite.  The  foregoing  have  been  given  as  specimens,  from  which  it 
is  hoped  that  the  student  will  be  able  to  learn  how  the  equations  of 
loci  referred  to  rectangular  axes  are  produced  from  the  definitions  of 
the  loci.  A  great  many  kinds  of  curves  are  suggested  by  the  study 
of  the  Properties  of  Curves.  Some  of  these 
win  be  noticed  hereafter.  Again^  Mechan- 
ics and  other  branches  of  Physics,  give  rise 
to  the  study  of  curves,  the  production  of 
whose  equations  requires  a  knowledge  of 
the  principles  of  Natural  Philosophy. 
Thus,  the  Tractrix  is  the  path  described  by  a 
weight,  W,  Fig.  72,  to  which  a  cord,  AW,  is  attached,  and  the  extrem- 
ity, A,  of  the  cord,  made  to  pass  over  the  path,  AB,  friction  being 
supposed  uniform  and  perfect.  Again,  the  Catenary  is  the  line  which 
a  perfectly  flexible  chain  assumes  when  its  ends  are  fastened  at  two 
points,  as  A  and  B,  Fig.  73,  nearer  together         Ak  ^B 

than  the  length  of  the  chain.  Caustics  are  an 
interesting  class  of  curves  consequent  upon  the 
laws  of  reflected  light.  The  next  chapter  gives 
several  varieties  of  curves  caUed  Spirals. 


Fig.  73. 


CHAPTER  II. 

THE  METHOD  OF  POLAR  CO-ORDINATES. 


SJECTIOJSr  I. 
Of  the  Point  in  a  Plane. 

100.  JPvop. — The  Position  of  any  Point  in  a  plane  can  be  designated  j 
hy  giving  its  Distance  and  Direction  from  a  fixed  point  in  the  plane.     In 
order  to  indicate  direction,  a  fixed  line  has  to  be  assumed. 

III.— Let  A,  Fig.  74,  be  the  fixed  point,  and 
AX  the  fixed  line.  Let  r  represent  the  distance 
from  the  fixed  point  to  the  point  to  be  desig- 
nated, as  AP,  AP',  etc.,  and  6  the  angle  in- 
cluded between  the  fixed  line  and  the  line  from 

the  fixed  point  to  the  point  to  be  designated,  as        '  "       '  P'" 

PAX,  P' AX,  etc.,  etc.     It  is  evident  that  by  Pig.  74. 

giving  all  possible  values  to  0,  and  r,  all  points  in  the  plane  of  the  paper  may  be 
located.  Thus,  for  P,  we  have  0  =35o,  r  =  5  ;  for  P',  0  =  1200,  r  =  10  •  P'' 
0=195°  r=  8  ;  and  for  P",  0=3450,  r  =  11. 

101*  Def's. — TheJPole  is  the  assumed  fixed  point,  as  A.  The 
JPrime  Madius  (called  also  the  Initial  Line,  and  the  Polar  Axis) 
is  the  assumed  fixed  hne,  as  AX.     The  Radius  vector  is  the 

distance  from  the  pole  to  the  point  to  be  designated,  as  A  P,  A  P' 
etc.  The  Variable  Angle  {O)  is  the  angle  indicating  the  di- 
rection of  the  point  from  the  pole.     When  6  is  reckoned  around  from 

right  to  left,  it  is  called  +  ;  when  reckoned  from  left  to  right, . 

The  radius  is  -f  when  estimated  in  the  direction  of  the  extremity  of 
the  arc  measuring  the  variable  angle ;  and  it  is  —  when  estimated  in 
the  opposite  direction,     r  and  0  are  the  Polar  Co-ordinates. 


102,  JProp, — The  Polar  equations  of  a  Point  are,  r  =  a,  and  0  =  b  ; 
since,  by  giving  suitable  values  to  r  and  d,  all  points  in  the  plane  can  be 
located. 

Ex.  1.  Locate  r  =  5,  0  =  \7t. 

Solution. — The  radius  being  1,  it  is  the  semi-circumference.  Hence  lit  =  6O0. 
Now,  lay  off  PAX  =  0  =  6O0,  Fig.  74  ;  and,  taking  AP  =  5,  P  is  the  required 
point. 


68 


THE  METHOD   OF  POLAB   CO-ORDINATES. 


Exs.  2  to  6.  Locate  r  =  3,0  =  ^7r:  r  =  4.,  d  =  ^7r:  r  =  6,  0. 
r  =  —6,  0  =  135°  :  r  =  4t,  6  =  —60°. 


^TT 


Ex.  7.  Show  that  r=lO,0 
^=135°. 


-45°,  is  the  same  point  as  r  =  — 10, 


103,  JPvob, — To  find  the  Distance  between  two  points  given  by  their 
polar"  co-ordinates. 

Solution. — Let  the  co-ordinates  of  P,  Fig.  75,  be 
(r,  6)  ;  and  of  P',  (r',  6').  We  are  to  find  PP'  in 
terms  of  r  r' ,  6,  and  9'.  Now,  in  the  triangle 
PAP',  AP  =  r,  AP'  =  r' and  the  included  angle 
PAP'  =  0  —  0'.  Hence,  representing  PP'  by  D,  we  ^ 
have  from  principles  of  trigonometry 

D  =  \/r'  -f-  1'"^  —  '^rf'  cos  (6  —  Q'),     q.  e.  d. 

Ex.  1.  Eind  the  distance  between  r  =  S,  d  =  \7t,  and  r  =  4,  d  =  ^7t. 
Ex.  2.  Find  the  distance  between  (8,  f  ;r),  and  (3,  \^7t). 

Ex.  3.  Find  the  distance  between  {\/\  45°),  and  (1,  0°). 

Results  in  the  last  three  examples,  not  in  order,  7,  1,  5, 


Fig.  75. 


■^♦» 


SUCTION  IL 
Of  the  Eight  Line. 

10 4z*  I^rob, — To  produce  the  Polar  Equation  of  the  Right  Line. 

Solution. — The  form  of  this  equation  (like  all  others)  depends  upon  the  con- 
stants assumed.     We  will  consider  two  forms. 

1st.  When  the  constants  are  the  length  of  the  perpendicular  from  the  pole  upon  the 
line,  and  the  angle  which  this  perpendicular  makes  with  the  prime  radius.  Thus  in 
Fig.  76,  let  M  N  be  any  line  ;  A,  the  pole  ;  AX,  the 
prime  radius  ;  the  perpendicular  from  the  pole  upon  the 
line,  A  D  :=  p  ;  and  the  angle  which  the  perpendicular 
makes  with  the  prime  radius,  DAX  =  a.  Let  P  be 
any  point  in  the  line  M  N ,  and  its  co-ordinates  be 
(r,  0).  Now,  in  the  right  angled  triangle  PAD,  we 
have  AD  =  A P  cos  PAD,  or  p  =  7-  cos  (9  —  a)  ; 
V 


r  = 


E.   D. 


cos  (9  —  a) 
2nd.    When  the  constants  are  the  intercept  on  the  prime 


Fig.  76. 


OF  THE   RIGHT   LINE.  69 

radius,  and  the  angle  which  the  line  makes  with  the 
prime  radius.  In  Fig.  77,  let  M  N  be  any  line 
referred  to  the  pole,  A,  and  the  prime  radius,  AX. 
Represent  the  intercept,  AT,  by  c,  and  the  angle 
NTX,  by  a.  Let  P  be  any  point  in  the  line, 
and  its  co-ordinates  AP  =  r,  and  PAX  =  0.  ^\ 
The  angle  XP  A  =  0  —  a  ;  and,  from  the  triangle 
PTA,  we  have  AP  :  AT  :  :sinPTA:siu  TPA, 
or  r  :  c  : :  sin  a  :  sin  i^Q  —  a)  ; 
sin  a 

.  • .     r  ==  —. t; c.       Q.  E.  D. 

sm  (6  —  a) 

P 
ScH.  1. — Discussion  of  the  equation  r  = .     When  9  =  0,  we  have 

"^  ^  cos  (9  — a) 

P 
r  =. — .     This  is  as  it  should  be,  for,  when  9  =  0,  r  =  AT,  Fig.  76, 

cos  ( OC) 

AD  p 

which,  from  the  trianffle  ADT,  is  seen  to  be ~ ,  or  ; ^,.     The 

.cos  DAT         cos( — cc) 

—  sign  of  (X.  indicates  that  the  radius  vector  falls  upon  the  opposite  side  of 

the  perpendicular  from  that  assumed  in  producing  the  equation.  .  .  .  When 


r=  a:,    r  =  -J—  =  p,  as  it  evidently  should.  .  .  .  When  9  —  ol  =  90°, 


r  =  -  =  00.     In  this  case  the  radius  vector  becomes  parallel  to  the  line, 

and  hence  oo .  .  .   .  From  9  —  a  =  90°  to  9  —  a  =.  270°,  r  is  negative,  as 

it  should  be ;  since,  in  order  to  reach  the  line  M  N ,  it  must  be  produced 

hackioard,  i.  e. ,  from  the  pole  in  a  direction  opposite  to  the  extremity  of  the 

■^     arc  9  measured  from  the  prime  radius  around  to  the  right.  .  .  .  From  9  — • 

K    a  =  270°  to  9  —  a  =  360°,  ?*  is  +  ;  and  at  9  —  a  =  360°,  r  =  p,  as  it  should. 

P     Also,   ait  9  =  360°,   r  = ^ =  AT.  .  .  .  When  the  line  M  N  passes 

cos( — a) 

through  the  pole,  r  = ,  which  is  0  for  all  values  of  9  except  9  =^ 

cos  (9  —  a) 

(90°  +  a),  for  which  r  =  -,  i.  e.,  indeterminate.  The  0  values  of  r  indi- 
cate that  we  have  not  to  pass  any  distance  from  the  pole  to  reach  the  line  ; 
and  the  -  value  indicates  that  for  all  values  of  r,  its  extremity  is  in  the  line 

MN-  .  •  .  Finally,  when  a  =:0,  the  line  MN  becomes  perpendicular  to  the 

P 


prime  radius,  and  its  equation  is  r  = 


cos  9 


ScH.  2. — Discussion  of  the  form  r  =:  -; — — .c.     For  9  =  0,  r  =  — c  ; 

sm  (9  —  a) 

which  is  evidently  correct,   as  it  is  reckoned   backward,  and  equals  c,   in 
length.  .  .  .  For  all  values  of  9  <;  a,  r  is  negative,  and  hence  is  reckoned  back- 


70  THE   METHOD   OP   POLAE    CO-OBDINATES. 

ward.  .  .  .  For  0  =  a,  y  = c  :=  oo,  as  it  should,  since  it  is  then  par- 
allel to  the  line.  .  .  .  For  values  between  6  >■  «:,  and  0  =  180°  -f  or,  r  is 
positive.   ...  At  0  =  180°  -\-  a.,  r  becomes  infinite  ;    and,  when  0  passes 

^  ^ .  .  .  sin  oc 

180°  -^  a,  r  IS  again  negative.  .  .  .  For  0  =  180°,   r  = c  =  c, 

sin  (180°  —  a) 

«.  e.,  AT.  .  .  .  If  the  hne  MN  passes  through  the  pole,  c  =  0,  whence 

sin  a  '  c  0 

r  =  ~ — ; r  =  -: — ; =  0,  for  all  values  of  0  except  Q  ^^a,  in  which 

sm  (0  —  a)       sin  (0  —  a}  ^ 

0 

case  r  =  -.     These  results  are  evidently  correct,  for  in  the  former  cases  we 

have  to  go  0  distance  from  the  pole  in  the  specified  directions,  in  order  to 
reach  the  Hne  ;  and,  in  the  latter  (when  6  =  a)  the  radius  vector  falHng  on 
the  line  MN,  its  extremity  is  equally  in  the  line  for  all  values  of  r. 

Ex.  1.  What  is  the  polar  equation  (first  form)  of  a  line  the  nearest 
point  in  vrhich  is  6  from  the  pole,  and  the  perpendicular  to  which 
makes  an  angle  of  45°  with  the  prime  radius?  Where  does  this  line 
cut  the  prime  radius  ?  For  what  values  of  ^  is  r  infinite  ?  What  is 
the  value  of  r  for  d=  75°  ?  For  d  =  15°  ?  Construct  these  values 
of  r  and  verify  them  by  drawing  the  line. 

Ex.  2.  What  is  the  polar  equation  (first  form)  of  a  right  line  per- 
pendicular to  the  prime  radius,  and  which  cuts  it  at  4  to  the  left  of 
the  pole  ?  What  is  the  value  of  r  when  ^  =  60°  ?  Why  is  the  sign 
of  r  negative  in  the  latter  case  ?  Between  what  values  of  6  is  r  posi- 
tive ?    What  is  the  value  of  r  when  0  is  120°  ? 

4 

The  equation  is  r  = . 

^  cos  d 

Ex.  3.  Give  the  equation  of  a  line  parallel  to  the  prime  radius,  and 

m 
10  above  it ;  also  at  m  below.       The  latter  equation  is,  r  == 


sin  6 


SECTION  III. 
Of  the  Circle. 


105,  I^roh, — To  produce  the  Polar  Equation  of  a   Circle,  the  pole 
being  in  the  circumference,  and  the  polar  axis  being  a  diameter. 


or  THE   CIRCLE. 


71 


Solution. — Lot  A,  Fig.  78,  be  the  pole  ;  B, 
the  radius  of  the  circle  ;  and  P,  any  point  in  the 
circumference.  Then  AP=r,  and  PAB^Q. 
Now,  from  the  right  angled  triangle  APB,  we 
have,  by  trigonometry,  r  =  2B  cos  0.     q.  e.  d. 

ScH.  1. — Discussion  of  ike  Equation.  If  0 
=  0,  r  =  2B.     If   6  =  ^7t,  or  f  tt,   r  =  0.     For  Fig.  78. 

values  of  0  between  j7t  and  f  tt,  r  is  negative,  indicating  that  for  these  val- 
ues of  0  the  radius  vector  must  be  produced  backward  to  meet  the  circum- 
ference. (The  student  should  observe  that  the  results  obtained  from  the 
equation,  accord  with,  the  values  as  observed  from  the  figure.) 


ScH.  2. — If  the  pole  is  at  the  centre,  the  equation  is  evidently  r  =  R,  for 
all  values  of  0. 


106.  IProh, — To  produce  the  General  Polar  Equation  of  a  Circle. 


Solution. —Let  the  constants  be  (r',  0'),  the 
co-ordinates  of  the  centre,  C,  Fig.  79  ;  and  E, 
the  radius  of  the  circle.  Let  P  be  any  point  in 
the  circumference,  and  its  co-ordinates,  A  P  and 
the  angle  P  AX ,  be  r  and  0.  Now,  since  the  dis- 
tance between  the  points  C  and  P  is  constant  (R), 


we  have  from  {103)B—\/r^-\-r''^ — 2rr'cos(0 — 0'). 
Whence  we  have, 

r-2  _  2r'  r  cos  {6  —  Q')  =  B^  — 


Fig.  79. 


0-    E.    D. 


ScH. — Discussion  of  the  Equation. — Solving  the  equation  for  r,  we  nave, 
7-  =  r'  cos  (0  —  0')  d=  \/i?^— r'^sin^  (0—0').  This  value  of  r  is  real  only  for 
such  values  of  0  as  render  r"^  sin2(0  —  0')  <or  =  R^ ;    i.  e.,  when  sin(0  —  0') 

R  R       PC 

is  numerically  <  or  =  —  .    Now,  sin  (0  —  0')  ^  sin  P  AC  =  —  = makes 

r'  r'       AC 

P  a  right  angle,  and  hence,  A  P  tangent  to  the  circle.    But  sin2(0  —  0') 


r"i 


R 


gives  sin  (0  —  0')  =  ±  -7- ;  hence  there  are  two  positions  of  r  in  which  it  is 

tangent  to  the  circle.     (This  is  the  familiar  truth  of  Geometry,  that  from 
any  point  without  a  circle  two  tangents  can  be  drawn  to  the  curve.)     The 

condition  sin  (0  —  0)  == ;-  is  satisfied  by  the  lower  point  of  tangency,  for 

which  0  —  0'  becomes  — ,  and  hence,  sin  (0  —  0')  is  — .     Between  the  limits 

R  R 

BID.  (0  —  0')  =  —;  and ;-,  there  are  two  real  and  unequal  values  of  r  for 


72  THE  METHOD  OF  POLAR  CO-OEDINATES. 

every  value  of  0,  as  evidently  should  be  the  case,  since  between  these  limits 

the  radius  vector  meets  the  curve  in  two  points ....  When  VR^ —  r''^&m\b b"\ 

<c^r'  cos  (9—6')  the  two  values  of  r  have  the  same  sign;  but  when 
y/ U-i  —  r'2smH6  — G')  >  ^' cos  (9  — 0'),  r  has  different  signs.  In  the 
former  case  the  i^ole  is  without  the  circle, — in  the  latter  within.  These 
facts  readily  appear  by  solving  the  inequality.  Thus,  in  the  latter, 
R  —  r'2  sin2(9  —  9')>  r'^  cos2(9  —  9'),  or  R  >  r'^  sin2(9  —  9')  +  r'^  cos2(9  —  9'). 
But  the  latter  member  reduces  to  r"^.  .'.  R^r',  which  puts  the  i^ole 
within  the  circle.*    (For  other  cases,  see  examples  below.) 

Ex.  1.  What  is  the  equation  of  a  circle  whose  radius  is  10,  and  the 
polar  co-ordinates  of  whose  centre  are  (15,  ^tt)  ?  What  values  of  0  in- 
dicate that  the  radius  vector  is  tangent  to  the  circle  ?  Between  what 
limits  of  ^  is  r  real  ?  Between  what  imaginary  ?  What  is  the  posi- 
tion, and  what  are  the  values  of  0  when  the  radius  vector  passes 
through  the  centre  ?     Construct  the  figure. 

The  equation  is  r-  —  30  sin  Or  =  —  125  ;  or  r  =  15  sin  0  ± 
\/lOO  —  225  cos*  0.  The  posilions  of  tangency  are  cos  ^  =  f , 
and  cos  0  =  —  |,  when  ?^  =  5v/5. 

Ex.  2.  Give  and  discuss  as  above  the  polar  equation  of  the  circle 
whose  centre  is  at  (8,  \7t),  and  whose  radias  is  10.  Does  this  circle 
cut  the  polar  axis ;  and,  if  so,  where  ?  How  do  you  determine  this 
jDoint  from  the  equation  ? 

Equation,  r^  —  8\/2  (sin  0  -{-  cos  0)r  =  oQ.     Condition  of  tangency, 
sin  ^  +  cos  <9  =  f  v^ — 2.     .'.  The  pole  u  within;  as  appeal's  also 

from  —  ^  1.     Cuts  the  polar  axis  £j^  ?-  =  (4  ±  \/34 )  v/  2. 

Ex.  3.  Show  that  the  polar  equation  of  a  circle  is  r^  —  2r'r  cos  0 
=  JR2  —  r'2  when  the  polar  axis  passes  through  the  centre  and  the  j)ole 
is  without  or  within  the  circle.  (Prove  this  directly  from  a  figure 
without  reference  to  the  preceding  forms. ) 

Ex.  4.  Deduce  from  the  general  form  in  100,  the  form  in  10 S, 
and  also  the  one  in  Ex.  3. 


R 

*  This  may  also  be  observed  from  sin  (Q  —  Q',)  =  +  —,   whicli  is  the  condition    of   tangency. 

r' 

This  is  possible  only  when  R  <;  r',  i.  e.,  when  the  pole  is  without  the  circle.     When  R  =r'  the  two 

tangents  become  one. 


OF   THE   CONIC   SECTIONS. 


73 


Ex.  5.  Let  the  student  show  from  the 
annexed  figure  that  in  the  equation  r 
=r'cos  [d  —  e')±iV Ri—r'-^sm^  {6—0^ 
the  rational  part,  r'cos  {6  —  0'),  is  the 
chord,  A  D,  of  the  circle  described  up- 
on A  C  (=  r')  as  a  diameter,  and  that 
the  radical  part,  \^Ii^ —  r'^  sin^  {0 — 0'),  '"^^^^        ^^^^^ 

is   P  D  =  P' D,  +  for  P  D  and  —  for 
P'D.     B  and  B',  are  the  points  where  the  radical  becomes  0. 


■»♦■»■ 


SUCTION'  IV. 
Of  the  Oonic  Sections. 

107 >  I^voh, — To  produce  the  Polar  Equation  of  a  Conic  Section. 


Solution. — Let  P  be  any  point  in  tlie  curve  ;  EC  the     e 
directrix  ;  A,  the  focus  and  pole  ;   and  AX,   the  axis  of 
the  curve  ana  the  polar  axis.  Let  2p  be  the  latus  rectum ;  e, 
Boscovich's  ratio,  whence   p=CAXe;     AP  =  r,   and 
PAX=0.       Then     AP  =  CD  X  e  =  (CA -f  AD)e. 

But     C  A    =  —  ,  and    A  D  =  r  cos  6 ;  whence,  r  = 

6 


( f-  r  cos  0  )e. 


P 


1  — e  cos 


Q.    E.    D. 


Fig.  80. 


108.  CoR  l.—The  JPolar  JEquation  of  the  I'arabola. 

Since  in  the  parabola  e  =  1,  its  polar  equation  is 

P  P 


1  —  cos  6 


,  or  r  == 


versin  0' 


109.  Cor.  %—The  JPolar  Equation  of  the  Mlipse  and 

Hyperbola  in  terms  of  the  semi-transverse  axis  and 

eccentricity.     Since  Boscovich's  ratio  (e)  and  the  eccentricity  are 

the  same  {48),  and,  numerically,   p  ==  A(l  —  e^),   this   equation   is 

Ajl  —  e^) 

1  —  e  cos  0' 


ScH.  1. — Discussion  of  the  Polar  Equaiicm  of  the  Parabola. — ^For  9  =  0, 


74 


THE   METHOD    OF   POLAR   CO-OKDINATES. 


P  ^  P 

^  ^  1  —  COS  e  ^®^°^®^  ^'  ^  flTT  =  '^>  *'•  ^•'  *^®  radius  vector  falling  upon 

the  axis  does  not  meet  the  curve For  any  value  of  6  >  0,  however  small, 

r  is  finite  ;  which  shows  that,  if  a  line  be  drawn  from  the  focus  making  any 
angle  however  small  with  the  axis  of  the  curve,  it  meets  the  curve  at  a  finite 
distance.  ....  For  ©  =  90°,  r  =  p,  as  it  should For  B  =  180°,  r  = 

P  P  m     • 

-. ,.  =  -.    .    -.  =  ip-     This  is  evidently  correct,  since  r  becomes  A  B, 

Fig.  80,  when  Q  =  180°.     But,  in  the  parabola,  AB  =  ip For  6  =  270°, 

r  =  p,  as  it  should.     (Let  the  student   discuss  in  like  manner  the  form 
P 


r  == 


versin  6 


) 


ScH.   2. — DiscussioTi.    of    the    equation    r  = 
u4(l  —  e2) 

:; for   the  Ellipse. — ^For  6  =  0,  r  = 

1  —  e  cos  9  -^ 

A{l  —  e^) 

— — — =  A(X  -{- e)  =  A  -\-  Ae )  which  makes 

AC  =  J.  +  Je,  as  it  should  {49) For  the 

point  P  at  the  extremity  of  the  conjugate  axis, 

T-r-      n^  „  AO         Ae 

J^ig.  81,    cos  6    =   — —  =  — .      Hence   r  = 


or   r 


AP 

-  Ae^   =  ^(1  —  e2). 


^(1  —  e2) 
Ae^  '■ 
r 
agreeably  to  [4=6] For  6  =  90 

be. 


l  +  e 


r  =  ^(1  —  e'^)  =  p For  0  = 

=  A  —  Ae,  which  is  the  value  of  A  B,  as  it  should 


ScH.  3. — Discussion  of  the  JEqua- 


A{1  —  e^ 


for  the  Hyper- 


tion  r 

1  —  e  cos 

bola. — Kemembering    that    in    the 

hyperbola  e  >  1,  we   observe    that 

^(1 — e-)  is  essentially  negative,  and 

hence   that  the   sign  of  r  depends 

upon  the  sign  of  the  denominator, 

1—e  cosQ ;  r  being  +  when  e  cos  e>l, 

and  —  when  ecose<l.      Now  for 

6  =  0,   r  =  __ ::   =  A  -^  Ae, 

1  — e 

which  indicates  that  A,  Fig.  82,  is  the  pole,  and  C  is  the  point  located 
when  6  =  0,  as  AC  =  yl  +  ^e  [49) As  6  increases  from  0,  cos  6  dimin- 
ishes, which  diminishes  the  numerical  value  of   1  —  e  cos  6,  till  6  =  cos-i- 

e 

(though  leaving  it  negative,  and  hence  r  positive),  hence  r  increases  and  the 


Fig.  82. 


case 


OF  THE   CONIC   SECTIONS.  75 

arc  C  M  is  traced When  1  —  e  cos  9  =  0,  i.  e. ,  when  cos  6 = -Yin  which 

e  \ 

6  =  cos  ^-\,  r  =  00.    In  this  position  r(  A  P')  becomes  parallel  to  the  asymptote 

OC       -4       1 

OT,  a  hneso  drawn  that  DC  =  B,  whence  cosDOC  == = —  ==_ 

OD      Ae      e 

When  0  passes  the  value  cos~^-,  1 —  e  cos  0  becomes  positive  and  renders  r  nega- 
tive, and  the  left  hand  branch  begins  to  be  traced,  the  point  moving  in  the  di- 
rection M'  B  M  ".    Thus,  when  9  =  ZZ"AX,  r  being  negative  is  reckoned  back- 

A{1 eA 

wards  and  the  point  P"  is  located.     When  0  =  90°,  r  =  — ^ =  —  ^9,  and 

P'"  is  located,  AP'"  being  equal  to  — p.     As  0  passes  from  90°  to  180°,  the 

arc  P'" B  is  traced.     At  0  =  180°,  r  =      ^    ~^  =  ^^^~ -^  =  A  —  Ae  = 

1  — e(  — 1)  1  +  e 

—  A  B,  as  it  should.     (In  like  manner  let  the  student  observe  that  as  0  passes 

from  180°  to  270°,  BP^^  is  traced ;   at  0  =  270°,  r  =  —  p  =  AP^^  ;    from 

0  =  270°  to  0  =  cos~^-  m  the  4:th  quadrant,  r  remains  negative  and  P'^  M"  is 

traced ;  at  0  =  cos~^-  in  the  4:th  quadrant  r  =  00  =  A  P^,  parallel  to  the 

e 

asymptote  OT'  ;  and  that,  finally,  from  this  value  of  0  to  0  =  360°,  r  be- 
comes positive  again,  and  the  arc  M  '  C   is  traced. ) 

Ex.  1.  What  is  the  polar  equation  of  a  parabola  whose  principal 
parameter  is  8,  the  pole  being  at  the  focus  ?  What  is  the  length  of 
the  radius  vector  for  <?  =  60°  ? 

4 

The  equation  is  r  =  :; .    For  6  =  60°,  r  =  8. 

1  —  cos  0 

Ex.  2.  WTiat  is  the  polar  equation  of  an  ellipse  whose  axes  are  12 
and  8,  the  pole  being  at  the  focus  ?  What  are  the  focal  distances  ; 
i.  e.,  what  are  the  respective  values  of  r  for  d  =  0,  and  for  0  =  180°  ? 
What  is  the  semi-latus  rectum  ;  i.  e.,  what  is  the  value  of  r  when  d 
=  90°? 

Q 

The  equation  is  r  = — .     The  focal  distances  are   6  4- 

3  —  \/5  cos  ^ 

2v  5,  and  6  — 2V  5.     The  semi-latus  rectum  is  2f. 

Ex.  3.  What  is  the  polar  equation  of  an  hyperbola  whose  trans- 
verse axis  is  6,  and  the  distance  between  the  foci  10  ?  What  is  the 
value  of  r  when  0  =  0?    When  0  =  90°  ?    W^en  0  =  180°  ?     What 

is  the  value  of  0  when  r  =  go  ? 
/  IP 

The  equation  is  r  = ^.     For  r=  00,  ^  =  53°8',  nearly. 

/  5  cos  0  —  3 


76  THE  METHOD  OP  POLAR   CO-ORDINATES. 

Ex.  4.  Show  that   the  polar  equation  of  the  parabola  becomes 

^__  £. — __  when  the  prime  radius  lies  in  the  position  AC,  Fiq. 

1  +  cos  d'  '■ 

80,  and  ^' is  reckoned  from  it  around  to  the  left  (in  the  usual  direc- 
tion). Show  that  in  this  form,  for  ^'  =  0,  r  =  \p  ;  and  for  0  =  180*^, 
r*  =  00 . 

SuG. — This  form  is  deduced  from  r  = ;:,  by  substituting  for  6,  6' — 180°, 

1  —  cos  6 

as  the  initial  line  is  revolved  forward  180°. 

Ex.  5.  A  comet  is  moving  in  a  parabolic  orbit  around  the  sun  at 
its  focus,  and  when  at  100,000,000  miles  from  the  sun,  the  radius  vec- 
tor makes  an  angle  with  the  axis  of  the  orbit  of  60°.  What  is  the 
polar  equation  of  the  comet's  orbit  ?  How  near  does  it  approach  the 
sun?    How  does  it  appear  in  a  parabola  that  r  =  2p,  for  0  =  60°  ? 

,_,  ^.      .  50,000,000 

I  fie  equation  is  r=  — -. 

^  1  —  cos  ^ 


■4»»- 


SECTION  V, 
Of  Higher  Plane  Curves. 

[Note. — All  of  this  section,  except  the  portion  upon  Spirals,  maybe  omitted  in  a  shorter  course, 
if  thought  desirable.] 


110,  JPTOh,     To  produce  the  Polar  Equation   of  the  Cissoid  of 
Diodes. 

Solution. — Let  AM,  Fig.  59,  be  the  cissoid,  A,  the  pole,  AX,  the  polar  axis, 
and  P  any  point  in  the  curve  ;  whence  A  P  =  r,  and  P  AX  =  6.     Let  A  B  =  2a. 

TVT  A   r-.  *    P^  f,  r^'r-,  n         "D     ^    r^ '    r-,  E'B^  (2a     SlU    S)-' 

Now  AP=r=  AD  sec  0=  D  B  X  sec  6.     But  D    B=— — -— = 

AB  2a 

=  2a  sin^  6.     Therefore,  r  =  2a  sin^  6  sec  0,  or  r  =  2a  sin  0  tan  0.  q.  e.  d. 

ScH. — Discussion  of  the  Equation.  For  0  =  0,  r  =  0. . .  .For  6  =  45°,  r  = 
a\/2  ;  hence  the  curve  passes  through  C,  Fig.  59, . .  .For  0  =  90°,  r  =  oo. 
. . .  .For  0>9O°  and  <;270°,  r  is  negative,  and  the  branch,  AM',  is  traced 
while  0  is  passing  from  90°  to  180°,  and  the  branch,  A  M,  is  traced  a  second 
time  by  the  negative  radius  vector  while  0  passes  from  180° to  270°. . .  .While 
0  passes  from  270°  to  360°,  r  is  positive  and  A  M '  is  traced  a  second  time. 
Therefore  the  curve  is  traced  twice  by  one  revolution  of  the  radius  vector. 


OF  HIGHEK  PLAKE  CUKVES.  77 

111,    ^Tob,     To  produce  the  Polar  Equation  of  the  Conchoid  of 
Nicomedes. 

Solution. — In  Fig.  62,  draw  F  K  perpendicular  to  OD.  Let  O  be  tlie  pole, 
and  O  D  parallel  to  AX,  the  polar  axis.  Let  AO  =6,  and  A  B  =  a.  Then  P 
being  any  point  in  the  curve,  we  have  OP=r  =  OF-{-FP  =  OF-f-«.  But 
O  F  =  F  K  X  cosec  FO  K  =  6  cosec  0.     Therefore,  r  =  b  cosec  0  -\-  a.    q.  e.  d. 

ScH. — Discussion  of  the  Equation.      For  0  =  0,  ?•  =  co  ....  For  0  =  90°, 

r  =  b-\-a For  6  =  180°,  r  =  oo For  0  >  180°  and  <  360°,  cosec  9 

is  negative,    and   the   lower   branch  is  traced.      The  student  should  be 
careful  to  observe  the  several  forms,  as  when  a'^  b,  a  =  b,  a<^b.     (See 

85,) 


112,  JPvob,     To  produce  the  Polar  Equation  of  the  Lemniscate  of 
Bernouilli. 

Solution. — Using  the  same  notation  as  in  Fig.  68,  let  A  be  the  pole,  and  AX 
the  polar  axis.  Let  P  be  any  point  in  the  curve,  and  draw  A  P.  Then  A  P  =  »*> 
and  P  AX  =  6.     The  following  is  an  outline  of  the  solution  : 


F'  P'  =  c2  +  r^  -f  2cr  cos  B.   (1). 

FP"^  =  0*2  4-  r2  —  2cr  cos  0.  (2). 
Multiplying  (1)  and   (2)  together,   and  remembering  that   F'PX  FP  =  c2,  we 
have,  after  a  little  reduction  : 

r2  =  4.G^  cos2  0  —  2c2  =  2c2(2  cos2  0  —  1). 
But  2cos2  0  —  1  =  cos  20.     .  • .  r2  =  2c2  cos  20.    q.  e.  d. 

ScH. — The  pupil  should  discuss  this  equation  as  the  preceding  have  been. 

[Note. — It  is  frequently  more  convenient  to  obtain  the  polar  equation  of  a  curve  by  transforming 
its  rectilinear  equation,  according  to  a  process  to  be  explained  in  a  subsequent  chapter.  We  will 
close  tb.is  chapter  with  some  account  of  Spirals,  a  class  of  curves  of  much  historic  interest  ia  con- 
sequence of  the  labor  bestowed  upon  some  of  them  by  the  old  geometricians,  and  to  which  the 
method  of  polar  co-ordinates  is  specially  adapted.] 


i 


OF  PLANE  SPIRALS. 

113.  Def's. — A.  I*lane  Spiral  is  the  locus  of  a  point  revolv- 
ing about  a  fixed  point,  and  continually  receding  from  it  in  such  a 
manner  that  the  radius  vector  is  a  function  of  the  variable  angle. 
Such  a  curve  may  cut  a  right  line  in  an  infinite  number  of  points, 
which  would  render  its  rectiUnear  equation  of  an  infinite  degree. 
Hence,  these  loci  are  transcendental. 

The  Measuring  Circle  is  the  circle  whose  radius  is  the  ra- 
dius vector  at  the  end  of  one  revolution  of  the  generating  point  in  the 
positive  direction. 

A.  Spire  is  the  portion  generated  by  any  one  revolution  of  the 
generating  point. 


78 


THE   METHOD   OF   POLAR   CO-ORDINATES. 


114.  The  Spiral  of  Archimedes  is  the  locus  of  a  point 
revolving  around  and  receding  from  a  fixed  point  so  that  the  ratio  of 
the  radius  vector  to  the  angle  through  which  it  has  moved  from  the 
polar  axis,  is  constant. 


JProb,     To  construct  the  Spiral  of  Archimedes. 

Fig.    83,  be  the 


lis. 

Solution. — ^Let  A? 
pole,  and  AX  the  prime  radius. 
Through  A  draw  any  convenient  num- 
ber of  indefinite  radial  lines  (say  8) 
making  equal  angles  with  each  other. 
Since  6  and  r  are  to  vary  alike,  when  0 
=  0,  r=  0,  and  the  spiral  begins  at  the 
pole.  Take  any  distance,  as  Al,  on 
Aa,  twice  this  distance,  as  A 2,  on  Al*, 
three  times  the  same  distance,  as  A3  on 

Ac,  etc.,  etc.     Then  will  1,  2,  3 

17  be  points  in  the  spiral,    q.  e.  d. 

III. — The  dotted  Hne  ah  cdefg  is  the  measuring  circle,  and  Al  2  3 
the  first  spire.     8  9  10  11 15  16  is  the  second  spire. 

ScH. — The  several  spires  of  this  spiral  are  such  that,  the  distance  between 
any  two  consecutive  ones  measured  on  the  radius  vector  is  the  same,  and 
equal  at  all  points  to  the  radius  of  the  measuring  circle. 


7  8  is 


110.     JProb.     To  produce  the  equation  of  the  Spiral  of  Archimedes. 

Solution. — Letting  a  be  the  ratio  of  the  radius  vector  to  the  variable  angle,  we 

have  r  =  aO.     Or,   otherwise,   calling  the  radius  of  the  measuring    circle,    A  8, 

Fig.  83,  1,  for  this  value  of  r,  6  =  27t.     Let  3  be  any  point  in  the  curve,  whence  A3 

represents  r,  and  SAX,  or  Sahc  =  Q.     Now  from  the  definition,  r  :  1  :  :  6  :  27t. 

B 

.  • .    r  =    r— .     Q.  E.  D. 

27t 

117.  Cor. — Tlie   Reciprocal   or   Hyperbolic   Spii^al. 

This  Spiral  is  naturally  suggested  by  the  Spiral  of  Archimedes,  as  in  it  the 
radius  vector  varies  inversely  as  the  variable  angle.     Hence  the  equation  is 
lit 

CoNSTBUCTioN. — To  construct  theBecip- 
rocal  Spiral,  let  A  be  the  pole,  and  AX 
the  polar  axis.  Draw  any  convenient 
number  of  radial  lines  through  the  pole, 
making  equal  angles  with  each  other. 
Take  Al  any  convenient  length,  A2  = 
^Al,  A3  =  iAl,  A4  =  iAl,  etc.,  etc. 
The  points  123  4 8  —  B  are  points 


Fig.  84. 


OF  HIGHER   PLANE   CURVES. 


79 


in  th6  curve.  Since  r  can  become  0  only  when  6  =  go,  this  curve  continues  to  ap- 
proach the  pole  as  the  radius  vector  revolves,  but  reaches  it  only  upon  an  infinite 
number  of  revolutions. 


lis.  The  Lituus. — The  equation 
of  this  spiral  is  r  =  — ^-.     Let   the   stu- 

dent  construct   it  and  give  the  formal  Fig.  85. 

definition.     The  form  of  the  curve  is  given  in  Fig.  85. 

119.  TJie  LogarithTnic  Spiral. — In  this  spiral  the  radius 
vector  increases  in  a  geometrical  ratio,  while  the  variable  angle  in- 
creases in  an  arithmetical  ratio.  The  equation  is,  therefore,  r=  a9. 
If  a  be  the  base  of  a  system  of  logarithms,  this  equation  becomes  0  = 
log  r. 

CoNSTEUCTioN. — To  coustruct  r=a^,  let 
a  =  2.  Then  for  0  =  0,  r  =  1,  which  gives 
the  point  0.  For  0=1,  i.  e.,  the  arc  of 
*57.3o  nearly,  r=2^=2,  which  gives  the 
point  1.  For  6  =  2,  I  e.,  the  arc  of  114. 6° 
nearly,  r  =  2^  =  4:,  which  gives  .  the  point 
2.  As  0  increases  r  increases  much 
more  rapidly,  so  that  with  this  small  base 
(a  =:  2),  at  the  end  of  the  first  revolution, 
when  6  =  6.28  +  ,  r=  26-28+=  more  than  64. 
Heuce,  at  one  revolution  the  radius  vector 
would  be  64  times  Ao.  But,  though  r  in- 
creases so  very  rapidly,  it  is  easy  to  see  that 
it  does  not  become  oo  till  6  =  oo .  Again, 
letting  the  radius  vector  revolve  in  the  nega- 
tive direction  from  AX,  so  that  6  is  negative, 
we  have  for  0  =  —  1  =  OAa,  r  =  Aa  =  2  — ^ 
=  h  For  6  =  —  2,  r  =  2-2  =  1  Thus,  it 
appears  that  as  the  radius  vector  revolves  in  this  direction  it  generates  a  portion  of 
the  spiral  which  at  first  rapidly  approaches  the  pole,  but  cannot  reach  it  till 
6  =  cx)  ....  Were  we  to  take  a  =  10,  the  base  of  the  common  system  of  logarithms, 
the  change  of  r  would  be  so  rapid  that  we  could  represent  but  a  small  arc  of  tbo 
curve. 


OHAPTEE  m. 


TRANSFORMATION  OF  CO-ORDINATES. 


SECTION  L 
Methods  of  Passing  from  one  Set  of  Eectilinear  Axes  to  Another. 

120.  T>EF's.—Transfor7naHon  of  Co-ordinates  is  the 

process  of  changing  the  reference  of  a  locus  from  one  set  of  axes  to 
another,  or  from  one  system  of  co-ordinates  to  another.  The  prob- 
lem presents  itself  under  two  different  aspects  which  are  nearly  the 
converse  of  each  other  :  1st,  Having  given  the  equation  of  a  locus  re- 
ferred to  one  set  of  axes,  or  system  of  co-ordinates,  to  find  the  equa-- 
tion  of  the  same  locus  when  referred  to  another  set  or  system.  2nd, 
Having  given  the  equation  of  a  locus  referred  to  some  known 
axes,  to  find  the  position  of  a  new  set,  to  which,  when  the  locus  is  re- 
ferred, its  equation  will  take  some  specified  form.  The  axes,  or  sys- 
tem, to  which  reference  is  made  in  the  given  equation,  may  be  called 
the  Oldf  or  Primitive,  Axes  or  System,  and  those  to  which 
the  transformation  is  made,  the  Wew. 


Ill's. — The  equations  ic2 -j- 2/^  =  25,  y'^ 
=  .^(10  — x),  and  x(x  —  4)  -f-  ViV  —  6) 
=  12,  may  all  be  considered  as  equations 
of  the  same  locus  M  O  N ,  Fig.  87,  but 
referred,  respectively,  to  the  three  pairs 
of  axes  X,Xi',  YiY,';  XiX,', 
Y2Y,'  ;X.3X,',  Y.3Y3'.  Now,  having 
given  any  one  of  these  equations  any  oth- 
er of  them  niay  be  found  if  we  know  the 
position  of  the  new  axis  with  reference  to 
the  old.     The  process  is  transformation. 

Again,  we  are  familiar  with  various 
methods  of  designating  particular  points  Fig.  87. 

on  the  earth's  surface,  as  by  latitude  and  longitude,  or  by  their  distances  and  direc- 
tions from  a  given  point.  For  example,  we  may  give  the  position  of  Chicago  by 
stating  its  latitude  and  longitude  with  reference  to  the  meridian  of  Washington,  or 


Y^ 

0/- 

Y3 

Y. 

x;     A2 

V 

A, 

A3 

)  ^ 

X'3 

Y3' 

y     xa 

Y/ 

FROM   ONE   EECTILINEAR   SYSTEM   TO   ANOTHER. 


81 


by  giving  its  distances  from  the  tropic  of  Cancer  and  the  meridian  of  Greenwich, 
Eng. ,  or,  in  still  another  way,  by  giving  its  distance  and  direction  from  New  York 
city.  The  process  of  converting  one  of  these  descriptions  into  any  other  of  them, 
would  furnish  an  analogy  to  the  process  of  transformation  of  co-ordinates.  The 
first  two  descriptions  (considering  the  earth's  surface  a  plane),  would  be  equations 
of  a  point  (Chicago)  referred  to  rectangular  co-ordinates  ;  the  last  would  be  an  ex- 
ample of  polar  co-ordinates . 

"We  will  give  one  more  illustration,  as  it  is  of  the  highest  importance  that  the  na- 
ture of  the  problem  be  understood  from  the  outset.  Let  the  student  construct  a 
pair  of  rectangular  axes,  XiXi', 
Y 1 Y  J ',  with  the  origin  A  i ,  and  an- 
other pair,  X2X2',  Y2Y 2',  also  rect- 
angular, with  the  origin  at  A  2  [the 
point  (0,  — 1)  when  referred  to  the 
first  axes],  and  the  new  axis  of 
X,  X2X2',  making  an  angle  with 
the  primitive  axis  of  —  45°,  and  the 
new  axis  of  y  making  an  angle  of 
450.  Now,  upon  the  first  axes,  con- 
struct ic2  —  6xy  -\-  y-  —  6a;  -f-  2?/  -j-  5 
=  0,  and  upon  the  second  axes  con- 
struct 2/'  —  2a;-  =  2,  when  the  two  j  y 
equations  will  be  found  to  give  the  Fig.  88. 
same  locus.  The  problem  of  transformation  which  affords  this  illustration  may 
be  stated  thus  ;  To  transform  x-  —  6xy  -{-  y-  —  6x  -{-  2?/  -|-  5  =  0,  to  a  new  system 
of  rectangular  axes  having  the  new  origin  at  (0,  — 1),  and  the  new  axis  of  x  mak- 
ing an  angle  with  the  primitive  whose  tangent  is  — 1. 

To  illustrate  the  second  form  under  which  the  problem  of  transformation  pre- 
sents itself,  the  example  of  the  last  paragraph  may  be  stated, — Having  given  the 
equation  x'^  —  Qxy  +  2/^  —  6x  -{-  2y  -(-  5  =.  0,  as  the  equation  of  a  locus  referred  to 
rectangular  axes,  required  to  find  the  position  of  a  new  pair  of  axes  to  which,  when 
the  locus  is  referred,  its  equation  will  involve  no  terms  in  the  first  power,  or  in  the 
rectangle  of  the  variables.  The  result  of  the  solution  of  this  problem  would  be  the 
determination  of  the  position  of  new  axes  as  given  in  the  paragraph  above. 


ScH. — As  tliGfo7'7n  (not  the  degree)  of  the  equation  of  a  locus,  depends  in 
a  large  measure  upon  the  situation  of  the  axes,  or  upon  the  system  used,  it 
will  be  readily  seen  that  a  set  of  axes  in  some  particular  position,  or  some 
particular  system  of  co-ordinates,  may  be  best  suited  to  one  class  of  pi'ob- 
lems,  or  of  loci,  and  another  set  or  system  to  another  class.  It  is  therefore  de- 
sirable to  be  able  to  pass  at  will  from  any  one  set  or  system  to  any  other. 
This  transformation  is  effected  by  finding  the  values  of  the  co-ordinates  in 
the  ^ven  equation  in  terms  of  new  co-ordinates  (and  certain  constants)  and 
substituting  the  latter  for  the  former.  The  methods  of  doing  this  we  will 
now  explain. 


82 


TKANSFORMATION   OF   CO-OSDIXATES 


122,  JProh. — To  produce  the  general  formuloe  for  passing  from  one 
set  of  rectilinear  co-ordinates  to  another. 

SoiiUTioN.— Let  P,  Fig.  89,  be  any 
point  in  a  locus  M  N  referred  to  the 
Frimitive  Axes  A  i  X , ,  A  i  Y  ^ ,  the  co-or- 
dinates being  A  i  D  =  ^,  and  PD=:y. 
Let  A2X2,  A2Y2  be  the  New  Axes, 
the  CO  ordinates  of  the  point  P,  when 
referred  to  them  being  A2D'=a?2, 
and  P  D  '=y2 .  Let  the  angle  included 
between  the  primitive  axes,  Yi  A^Xi, 
be  /3  ;  the  angle  which  the  new  axis  of 
X  makes  with  the  primitive,    X2  IXi, 

be  a ;  the  angle  which  the  new  axis  of  y  makes  with  the  primitive  axis  of  x, 
Y2l'Xi  be  a  ;  and  the  co-ordinates  of  the  new  origin,  Aj,  be  AiG  =m,  AgG 
=:  n.  The  problem  now  is,  to  find  the  values  of  the  primitive  co-ordinates  x,  y,  in 
terms  of  the  new  co-ordinates  x^,  2/2,  and  the  constants  m,  n,  a,  a  ,  and  /?,  so  that 
the  latter  may  be  substituted  for  the  former  in  the  equation  of  a  locus  referred  to 
the  primitive  axes,  and  the  equation  be  thus  transformed  and  made  to  represent 
the  same  locus  in  terms  of  the  new  co-ordinates  ;  i.  e.,  referred  to  the  new  axes. 

We  have  a;  =:  AiD  =  AiG  +  AjE  +  D'F.  But  AiG  =  m  ;  and  from  the 
triangle  AjD'E,  AgE  :  ^-2  ••  sinAjD'E  :  sinAgED',  which  becomes 
A2E  :  X2  :  :  sin{/3 —  a)  :  sin /3,  since  AsD'E  =  D'ER  —  D'AgE  =  /3 — a, 
and  sin  D'EAg  =  sin  D'ER  =  sin/5  (the  sine  of  an  angle  equals  the  sine  of 

Xz  sin  (/5  —  a) 


Fig.  89. 


its  supplement).     From  this  proportion,    A2E  = 


manner,  from  the  triangle  PD'F,  we  have  D'F  = 

these  equivalents  in  the  value  of  x  as  given  above 

^2  sin  (/5  —  a)  -\~  y^  sin  (/3 


sin  p 
2/2  sin  (/5— a') 


Again,  y  =  P  D  =  A  2  G 


siny^ 
a') 


In  the  same 


Substituting 


D'E  -f  PF. 


(1). 


angles   A2D'E,  and   PD'F,  we  have  as  before   D'E  = 


But  A2G  =  n,  and  from  the  tri- 
x^  sin  a 


sin/i  ' 


and  P  F  == 


?/,  sm  a 
sin/i 


Hence 


y  =  n-\- 


X2  sin  a  -\~  y^  sin  a 
sin/i  ' 


(2). 


123 •  Cor.  1. — When  the  New  Axes  are  parallel  to  the  Primitive,  the 
formulcB  become 

X  =  m  +  X2,  (1);  and  y  =  n  +  ja,  (2); 
since  in  such  a  case  a  =  0,  and  a'==  p  ;  v)hence  sin  /^  =  0.  sin  a'  =^ 
sin  /?,  sin  (/?  —  a)=^  sin  /?,   and  sin  (/?  —  a')  =  sin  0  =  0. 


FllOM   ONE   BECTILINEAR   SET   TO   ANOTHER.  83 

124:,  CoFw  2. — To  pass  from  rectangular  axes  to  oblique,  we  have 
X  =  m  +  Xa  COS  a  -\-  y^  cos  a',   (1) ; 
and  y  ==  n  +  Xa  sin  a  -}-  y^^  sin  a',     (2). 
These  follow  readily  from  the  general  formulse  by  observing  that  in  this 
case  P  =  90°;    whence   sin  (/? —  a)  =  cos  a,  sin  (/? —  <3^')  =  cos  a', 
and  sin  yj  =  1. 

J.2S,  CoE.  3. — To  pass  from  one  set  of  rectangular  axes  to  another  set 
also  rectangular,  but  not  parallel,  we  have 

X  =  ni  +  Xa  cos  a  —  j^  sin  a,  (1)  ; 
and  y  =  n  +  ^2  sin  a  +  J2  cos  a,   (2). 
To  deduce  these  f^om  the  general  formulae  {122 ),  observe  that  fi  =  90°, 
and  ix' —  az=z 90°,  or  a'  =  90°  +  ^/  whence  sin  f3  =  l,  sin  (/? —  ^)  = 
cos  <a:,  sin  (y5  —  <3f')  =  sin  (90°  —  90°  —  a)  =  sin  ( —  a)  =:r.  —  sin  a, 
and  sin  a'  =  sin  (90°  +  <^)  =  cos  a. 

120,  Cor.  4. — To  pass  from  oblique  axes  to  rectangular,  we  have 

Xosin(/5  —  ^)  — J2C0r(/?— a') 

X  =  m  H : — •- ,  (1)  ; 

sni  fJ 

X,  sin  «:  +  T"  cos  a:  ,^. 

y  =  11  +   -^ 3-2^ ,  2). 

sm/i 

In   this   case   the   a'  —  a  of  the  general  formula   becomes   90°,   or 

a'  =  90°  +  <:i: ;     whence     sin  {(3   —    a')   =  sin  (/i   —   90°  —  a) 

=  sin  {—  [90°  —  (/?  —  a)]}  =  —  sin  [90°  —  (/5  —  a-)]  =  — 

cos  (/?  —  Of),  and  sin  a'  =  sin  (90°  -\-  a)  =  cos  ar, 

127,  Cor.  5. —  When  the  origin  remains  the  same,  and  only  the  direc- 
tion of  the  axis  is  changed,  m  =  0,  and  n  =  0,  and  we  have, 

To  pass  from  one  oblique  set  to  another, 

j^asin  (/?  —  a)  4-  ^/asin  (/?  —  a') 


X 


sin/i 


(10 


x^  sin  a  -\-  y^  sin  a'  ... 

'-'  =  -^ — i:^ — '  (2') 

To  pass  from  rectangular  to  oblique  axes, 

X  =  x-i  cos  a  -f  ^2  cos  a',  (I2) 

y  =  x^  sin  <x  -f  2/2  sin  «' ;  (23) 

Topassfro7n  one  rectangular  set  to  another, 

X  =  x^  cos  a  —  ^2  sin  a,  (I3) 

y  =z  X2  sin  «:  +  2/2  cos  of ;  (23) 

To  pass  from  oblique  to  rectangular  axes, 

iTosin  (/]  —  a)  —  11/2  cos  (/?  —  a), 


X 


sin  /^ 


(1.) 


X.,  sin  Q'  4-  v^  cos  a  .^  . 


84  TRANSFOBMATION   OF  00-OKDINATES 

ScH.— In  (I4),  (24)  the  angles  involved  are  those  which  the  new  or  rec- 
tangular axes  make  with  the  i^rimiiive  or  obhque  axis  of  x.  It  is  some- 
times convenient  to  have  formulce  for  passing  from  obhque  to  rectangular 
axes,  in  which  the  angles  involved  shall  be  those  which  the  jyrimiiive  or 
oblique  axes  make  with  the  new  or  rectangular  axis  of  x.  Such  formulce 
may  be  deduced  from  the  general  formulce,  but  are  more  readily  obtained 
from  (I2),  (22),  as  follows  : 

Multiplying  (I2)  by  sin  ex.' ,         x  sin  oc'  :=^  x^  sin  a'  cos  oc  -\-  y^  cos  ex.'  sin  a', 
Multiplying  (22)  by  cos  a  ,         y  cos  a'  =  x^  cos  a'  sin  a  -\-  y^  cos  a!  sin  a\ 

Subtracting,  x  sin  a'  —  y  cos  a  =  x.2  (sin  a  cos  a  —  cos  a'  sin  a) ; 

Whence  since  sin  a'  cos  a  —  cos  a'  sin  a  =  sin  [a    —  a)  we  have 

X  sin  a'  —  y  cos  a' 

^2  =  ^-r-, ^ — •       (I5) 

sm  [a   —  a) 

In  like  manner  eliminating  Xz,  we  have 

y  cos  ex  —  X  sin  a       ,_  ^ 

y2  =       ■    ,  , r-     25 

sm  [a   —  a) 

[Note. — It  will  afford  tlie  student  an  excellent  geometrical  exercise  to  produce  each  of  the  above 
Bets  of  formulcB  directly  from  a  figure.  The  forms  in  corollaries  1  and  2  are  the  most  important, 
and  should  be  fixed  in  memory.] 

Ex.  1.  Assuming  Sx  -\-  5y  =  15  to  be  the  equation  of  a  right  line 
referred  to  rectangular  axes,  find  the  equation  of  the  same  line  re- 
ferred to  parallel  axes  whose  origin  is  at  (1,  2). 

The  equation  is  ^x^  +  ^y^  =  2. 

Ex.  2.  Assuming  2j7  +  3i/  =  6  to  be  the  equation  of  a  right  line  re- 
ferred to  rectangular  axes,  find  the  equation  of  the  same  line  referred 
to  parallel  axes  whose  origin  is  at  (1,  — 2).  Also  to  new  parallel 
axes  whose  origin  is  at  ( — 3,  — 4).  Also  to  new  parallel  axes  whose 
origin  is  ( — 6,  6).  Construct  the  given  equation  and  verify  the  new 
equations  by  constructing,  on  the  same  figure,  the  new  axes,  raid 
observing  the  position  of  the  line  as  referred  to  them.  Notice  where 
the  line  cuts  the  several  axes. 

The  equations  are,  1st,  2^2  f  ^Vi  =  10 ;  2nd,  2^3  +  8^/2  =  24 ;  and 
3rd,  2^0+ 3?/,=  0. 

QiTERY. — Why  do  not  the  coefficients  of  x  and  y  change  in  the  above  transfor- 
mations ? 

Ex.  3.  Construct  the  locus  ^2  —  ga;  + 1/2  +  61/  =  0,  upon  rectangular 
axes.  Then  transform  the  equation  by  passing  to  new  parallel  axes 
whose  origin  is  (4,  — 3),  and,  drawing  these  axes  on  the  same  figure, 
construct  the  new  equation  {x^^  +  y^^  =25)  with  reference  to  these 
axes,  observing  that  the  two  equations  give  the  same  locus. 

Ex.  4.  Given  4</-  +  9-^-  =  3G  as  the  equation  of  a  locus  referred  f  > 
rectangular  axes,  to  transform  to  new  axes  with  the  same  origin,  the 


FEOM    ONE    KECTILINEAR    BET   TO   ANOTHEE. 


85 


tangents  of  tlie  angles  wliich  the  new  axes  of  x  and  y  make  with  the 
primitive  axis  of  x,  being  3  and  — 3,  respectively.  Verify  by  a  con- 
struction. 

Sug's. — The  formulce  are  x  =  X2  cos  a  +  2/2  cos  a',  and  2/  =  cca  sin  a  +  2/2  ^i^  <^'. 
In  this  case  tan  a  =  S,  v/lience  sin  a  ^V^'ni^  ^^^  cos  a  =\/-^q.  Also  tan  <x'  =  —  3, 
whence  sin  a'  =  V-^o,  and  cos  a'  r=  —  >/j^(j.  Introducing  these  values,  the /onn- 
wZa?  become  x  =  \/-^u{x.2  —  2/2)5  ^'nd  y  =  v^tuK'^i  4"  1^2)' 

The  transformed  equation  is  ^x^^  +  ^^21/2  +  ^l/z'  =  ^^ . 

Ex.  5.  Given  xy  =  16  as  the  equation  of  a  locus  referred  to  rect- 
angular axes,  to  pass  to  new  rectangular  axes  with  the  same  origin, 
the  new  axis  of  x  making  an  angle  of  45°  with  the  primitive  axis 
of  X.  Equation,  x^  —  y^^  =  32. 

Ex.  6.  By  the  same  transformation  as  in  Ex.  5,  show  that  y*  -\-  x*-\- 
Gx^yi  =  2,  becomes  ^2^  +  2/2'*  =  1-  Construct  the  locus,  and  both  sets 
of  axes,  and  observe  the  position  of  the  locus  with  respect  to  the  two 
sets  of  axes. 

Ex.  7.  Given  the  equation  y  =  ax  -\-h  (the  common  equation  of  the 
straight  line),  to  pass  to  oblique  axes  with  the  same  origin. 

Sug's. — The  formulce  for  transforming  are  x  = 
X2  cos  a-\-p2  cos  a',  and  y  ==  x^  sin  n:  -f-  2/2  sin  a'. 
Substituting  and  reducing,  we  have 


y-i  = 


a  cos  a 


sm  a 


•,  ^2  + 


sm  oc  —  a  cos  a:  sm  ol  —  a  cos  oc 

which  is  the  equation  of  a  right  line  referred  to 

oblique  axes  making  any  angles  (a,  a')  with  the 

primitive  axis  of  x.     Now,  if  we  desire  simply  the 

form,  of  the  equation  of  a  right  hne  referred  to  ob- 

Hque  axes,  we  may  consider  the  new  axis  of  x  as 

coinciding  with  the  primitive,  Tig.  90,  and  let  the  new  axis  of  y  make  any  angle, 

as  y5,  with  this.     Then  a:  =  0,  and  a'  =  ft  ;  whence  the  equation  becomes 

a  .  1) 

Vi  =  ttttj Trrrr-Tp  ^2  + 


Fig.  90. 


sin  fi  —  a  cos  (3 
Again,  letting  the  angle  NT  A]  =  aj,  a  = 
sin  a^ 


sin  (5  —  a  cos  ft 
sin  ai 


cos  a-i 


whence 


sin/^  —  a  cos /J 


cos  a^ 


sm  OTi 


sm  a^ 


sin/J 


Also, 


sma, 
cos  0:1 


cos/? 


siu  ft  cos  (Xi  —  cos  ft  sin  cx.x 
h  cos  ax 


sin(/? — ai) 


sin/^ — a  con  ft       sin  (/? — aj 
:  sin  BCAi,   or  AiC  :  h  ::  sin  (90o  -j-  a^) 
6  sin  (90° -f- <^ I )  hcosai 


=  AiC,  since  AiC  :  Ai  B  : :  sinCBAL 


sin  (/i  —  a-i)  sin(yS — a-i)' 

calling  A 1 C  ^  5'.     (See  34,) 


sin  {ft  — 
Substituting,  we  have  2/2 


a-i)  ;  whence  AjC 
sin  a, 


sinift — OTi) 


X2  +  h'. 


86  TRAI^TSFOEMATION   C?   CO-ORDINATES 

Ex.  8;  Given  the  equation  y  =  ax-\-b,io  find  the  position  of  a  new 
set  of  axes  parallel  to  the  primitive,  to  which,  when  the  locus  is  re- 
ferred, its  equation  shall  have  no  absolute  term. 

Sug's. — Substituting  for  y,  y^  -\-  n,  and  for  x,  x.^  -f-  w,  we  have  2/2  =  <^a*2  +  ^'^ 
-{-6  —  n.  Now,  if  the  new  axes  can  be  so  situated  that  am,  -|-  &  —  n  =  0,  or  ?i  = 
a-m-Y'h^  the  condition  required  will  be  fulfilled.  But  n  and  m  are  the  co-ordinates 
cf  the  new  origin,  and  the  condition  n  =  am  -f-  &  (/i  and  m  being  co-ordinates) 
designates  a  point  in  the  line  y  =  ax  -{-I).  Hence  the  new  origin  is  to  be  in  the 
line  y  ^^  ax  -\-  h.     (See  33 y  Sch,  1.) 

Ex.  9.  Transform  A"y'^  +  B-x'^  =  A"B-^,  to  parallel  axes  with  the  new 
origin  at  ( — A,  0).  Also  to  parallel  axes  with  the  origin  at  {A,  0). 
Also  to  parallel  axes  with  the  origin  at  ( — m,  —  n).   (See  SS») 


Ex.  10.   Transform  A-y^  +  B-x^  =  A'B^,  to  oblique   axes  with  the 

B^ 
A^' 
terms  of  the  diameters  lying  on  the  new  axes. 


same  origin,  such  that  tan  a  tan  a'  = —^  and  obtain  the  result  in 


Sug's. — After  making  the  substitutions  and  collecting  terms,  we  have 

{A-sin^a  -f-  B-cos~a')y2  -  -{-  {A-sm"a  -\-  B'^cos"a)x2 ^  -{-  2{A-sma  sina'  -\~  B^cosacosa' ) 

^2^/2  =  A^B-,  (1),  which  is  the  general  equation  of  an  ellipse  referred  to  oblique 

axes,  the  origin  being  at  the  centre.     If  these  axes  are  so  situated  that  tan  a 

sin  a  sin  a'  B'     ,     .         .       ,        ^  ,       ^        -,  ,-,     . 

tan  a   = = -,  A'^  sm  a  sm  a   -\-  B-  cos  a  cos  a  =  0,  and  the  term 

cos  a  cos  a  A^ 

in    Xzy-z     disappears,    and    the    equation    is    (^2  sia^a'    -\-   B^  cos^ a')y2^    4- 
(J.2  sin2  a  -\-  B-  cos'^  a.)x.2"  =  A-B^.     This  is  the  equation  of  the  ellipse  referred 
to  the  axes  required,  but  it  is  not  in  the  terms  required^  it  is  the  equation  of  the 
ellipse,    as     BPC,    referred    to    the    diameters 
A,  Bg,  AxDg,  but  is  in  terms  of  the  semi-axes 
Aj  B,    AjG,   and  the   angles  a,  a',  which  the 
new  axes   make  with    the   primitive    axis  of  x. 
Thus,  P  being  any  point  in  the  curve,  A 1 E  rep- 
resents a-j,    and   RE  2/2.      Nov/ in  this  equation, 
4  when  2/2  =0,  Xs  becomes  Aj  B2.     Hence  calling 

Ai  B2  Aj,  we  have  ^.j-  ^=  - — -. -^ , 

A^Bin-^'a-^-B^cos^a 

A^B^ 
or  A^sin^  a  -\-  B-  cos^  a  =  — r^.     In  hke  manner  for  x^  =  0,  2/2  becomes  Ai  D^. 

Ai^ 

A^B^ 

Hence  calling  Ai  D,  B,,  we  have  B-,^  = ,  or  A^ sin^ a'  4- 

^         A^  sm^  a'  +  252  cos2  a'  ^ 

A^B^ 
B'cos-a'  =  -zr-T.     Substituting  these  values  of  the  coefficients  of  y^-,  and  iCg^, 
-"1 

A  2  7-^2  A  2  7?2 

we  have  —jj—y-z-  -\ j-rajj^  =  A'^B\     Finally,  dividing  by  A'^B'^,  and  clearing  of 

fractions,  A^^y^^  +  Bi^Xs^  =  A^^B^^     q.  e.  t>. 


FROM   ONE   RECTILINEAR   SET   TO   ANOTHER. 


87 


ScH. — ^Diameters  so  situated  as  to  make  tan  a  tan  a'  = —  are  Conju- 

gate  Dia7aeters,  as  will  appear  hereafter.  Hence  the  equation  of  the  ellipse  re- 
ferred to  conjugate  diameters,  and  in  terms  of  those  diameters,  is  of  the  same 
form  as  the  equation  of  the  curve  referred  to,  and  in  terms  of,  its  axes. 

I     Ex.  11.  Transform  A^y"^  —  B^x^  =  —  A^B^  to  an  oblique  system  with 

I  B^  . 

tho  same  oriofin,  such  that  tan  a  tan  «'  =  -r-,  and  obtain  the  result 

in  terms  of  the  diameters  lying  on  those  axes. 

Sitg's. — The  student  is  expected  to  recog- 
nize this  as  the  equation  of  the  hyperbola 
referred  to  its  own  axes.  The  transforma- 
tion is  in  all  respects  like  the  above,  except 
that  the  diameter  represented  by  B^  is  im- 
aginary, i.  e.,  does  not  meet  the  real 
branches  of  the  curve,  hence  we  call  AiDa* ' 

Byy/^^n.,  or  (A;^D2)2  =  —  B,^  The 
equation  soiight  is  Ai-y^^  —  ^I'^.-Cj- =  — 
A,^B,^ 


Fig.  92. 


ScH. — In  each  of  the  two  preceding  examples,  there  were  given,  the  equa- 
tion of  the  locus,  and  the  position  of  the  new  axes,  from  which  to  find  the 
form  of  the  new  equation.  The  converse  of  this  problem  is  important  ; 
i.  e.,  Given  the  equation  of  the  locus,  and  some  specified  form  of  its  equa- 
tion, io  find  the  position  of  the  new  axes.  Thus,  for  example, — The  origin 
remaining  the  same,  what  must  be  tho  position  of  oblique  axes,  to  which, 
when  the  eUipse  is  referred,  its  equation  will  take  the  same  form  as  the 
common  equation.  To  solve  this  problem,  we  first  transform  A'^y-  +  -S"a;2  = 
A'-B-^  to  obhque  axes,  as  in  Ex.  10,  and  obtain  the  form  (1)  in  the  sugges- 
tions. It  then  remains  to  determine  what  values  a  aad  a  must  have,  i.  e., 
how  the  new  axes  must  be  inclined,  to  make  the  equation  take  the  primi- 
tive form.  Now,  the  required  form  has  no  term  in  Xii/z,  hence  the  coeffi- 
cient of  .^2?/ '.  must  be  0,  that  is,  J.2  sin  a  sin  a  +  5-  cos  a  cos  a  =  0.     From 

;  in  (X  sin  a'  B^        ,  ■,  ,i    ,  . i 

this, =  tan  a:  tan  a'  = ;  whence  we  learn  that  the  new  axes 

cos  a  cos  a'  A^ 

must  be  so  situated  that  the  rectangle  of  the  tangents  of  the  angles  which 
they  make  with  the  primitive  axis  of  x  shall  be  —  — ,  in  other  words  they 
must  be  conjugate  diameters.     Putting  tho  resulting  equation  in  the  form 

1,  and  maldng  Xs  =  0, 


A  -  sin2  a'  4-  B^  cos^  a'  A~  sin-  a  4-  B-  cos"  a 

A-B^  ^'    +  AB^ 

and  2/3=0,  successively,  we  find  that  the  squares  of  the  new  semi-diameters  are 
J.2J52  A^B^ 


and  -- 


This  equation  and  these 


A^  sin2  a  +  B^  cos^  cc"  A^  sin2  a-\-  B'^  cos^  a 

values  refer  to  any  pair  of  conjugate  diameters,  as  wiU  appear  hereafter. 


88 


TRANSFORMATION   OF   CO-ORDINATES 


2pm =0. 


Ex.  12.  Find  the  position  of  oblique  axes 
to  which  when  y^  =  2px  is  referred,  the  equa- 
tion will  still  have  the  same  form. 

Sug's.— Passing  to  oblique  axes  in  general,  the  equa- 
tion becomes 
{n  +  a^asin  a:-}-2/2sin  ay  =  2p{m -{-Xscos  a  +  yzcosa'); 

or,  expanding  and  collecting  terms 

srQ2a:'2/2  ^ + 2sin  a  sin  a'a;22/2  +  siusa  a;,  2 -|- 2n  sin  a' j2/2  +  2n  sin  <a: 

— 2pcosa'l     — 2pcos.£t 

Now,  in  order  to  meet  the  conditions  of  the  problem, 

1st,  There  must  be  no  absolute  term  ;  hence  n^  —  2pm  =  0,  (1)  ; 

2nd,  There  must  be  no  term  in  y^  ;  hence  2n  sin  a  —  2p  cos  a  =  0,  (2) ; 

3rd,  There  must  be  no  term  in  x.^y^  ;  hence  2  sin  a  sin  a'  =  0,  (3)  ; 

4th,  There  must  be  no  term  in  iCg-;  hence  sin^  a  =  0,  (4). 

If  these  conditions  can  be  fulfilled,  and  we  can  discover  the  position  of  the  new 
axes  which  fulfills  them,  the  problem  is  solved.  We  observe  that  as  there  are  but 
four  conditions,  involving  four  arbitrary  constants,  a,  a  ,  m,  and  n,  these  condi- 
tions can  be  fulfilled.  The  first  condition,  n'^  —  2pm  =  0,  or  n'^  =  2pm,  requires 
the  new  origin  to  be  on  the  curve,  since  n  and  m  bear  the  relation  of  co-ordinates 

to  the  curve.    The  second  condition,  2n  sm  a  —2p  cos  a  =0,  or ;  =  tan  a  =-, 

cos  a  n 

requires  that  the  new  axis  of  y  make  an  angle  with  the  axis  of  the  curve,  whose 
tangent  is  p  divided  by  the  ordinate  (n)  of  the  new  origin,  which  is  on  the  curve. 
(This  makes  the  new  axis  of  j^  a  tangent  to  the  curve,  as  will  afterwards  appear.) 
The  third  condition,  2sin  asin  a'  =  0,  gives  sin  a  =  0,  since  sin  a'  is  not  0.  This 
requires  the  new  axis  of  x  to  be  parallel  to  the  primitive  axis  of  x  (the  axis  of  the 
curve),  and  makes  it  a  diameter.  The  fourth  condition  sin^  a  =  0,  is  fulfilled  by 
the  last,  and  hence  requires  no  further  attention.  If,  therefore,  the  curve  is  re- 
ferred to  any  diameter,  as  AzXs?  and  a  tangent  Aa  Ya  at  its  vertex,  the  equation 
becomes  sin^  a'y  2^  =  (^P  cos  a  —  2n  sin  a)X2  ;  or  since  sin  a  =  0,  and  cos  a  =  1, 

Vo^  =    — ^.^^>,  which  is  the  form  required.      Putting     .    ., 
^^  sin-^a'  sm^a 

2/2^  =  2P2X2. 

2p 
ScH. — The  equation  3/2^  = 


-  =  2p.2,  we  have 


-X2   leaves  the  problem   indeterminate, 
sin2  a' 

inasmuch  as  a'  is  a  function  of  3/2  ;    hence  the  new  origin  may  be  anyivhere 

on  the  curve.      In  reality  the  problem  furnished  four  arbitrary  constants 

and  required  but  three  conditions  (the  third  and  fourth  being  but  one)  ; 

hence  we  may  impose   another ;  that  is,  we  may  put  the  origin  where  we 

please  on  the  curve. 

Ex.  13.  To  transform  A^y^  —  B^x"^  =  —  A^B^  so  that  the  hyperbola 
shall  be  referred  to  its  asymptotes,  i.  e.,  to  the  produced  diagonals 
of  the  rectangle  drawn  on  the  axes  of  the  curve. 


FROM   ONE   RECTILINEAR   SET   TO   ANOTHER. 

Yi 


89 


K 


Stjg's. — Let  AjXi  and  A1.Y1  be  the 
primitive  axes,  and  the  asymptotes  A,X2, 
AtY2  be  the  new  axes.  As  usual,  let 
X1A1X2  =  —  oc,  and  XjAiYj  =  a'. 
Then,  since  CB  =  BD  =  jB,    AiB=J., 

Ai  C  =  A I  D  =  ^y A'  4-  B\  sin  a  =  — 
B  A  .       . 


B 


cos  a  = 


,  and  cos  a'  ^= 


A 


sm  a  = 


:,  the  — 


Fig  94. 


y/A^  4-  B-^  \/a^  +  B^ 

sign  being  given  to  the  value  of  sin  a.  since 
a  is  reckoned  around  the  angular  point  Ai  from  left  to  right,  and  sin  a  is  the  sine  of 
a  negative  arc  less  than  90°.     Putting  these  values  in  the  formulce  for  passing 

A 


from  rectangular  to  obUque  axes,  we  have  x  =  (x.^  +2/2) 
B 


,  and  y  == 


(2/2  —  ^2)- 


V  A-^  4-  B^ 


s/  A^  +  B^ 
Now,  substituting  these  values  of  x  and  y  in  the  equation 


to  be  transformed,  there  results 


(2/2 


cc2)2^2-B2  —  (ccj  -f  y^y-A^B^ 


whence  expanding  and  reducing  Xg^/a  = 


A'  -f  ii^J 


=  —  A^B-^ ; 


Since  4- —  is  constant, 


4       -  4 

we  may  represent  it  by  c,  and  write  the  equation  tCj^/a  =  c.     In  the  case  of  the 

A^ 
equilateral  hyperbola  c  =  —,  A  being  the  semi-axis. 

Ex.  14.  To  find  a  system  of  oblique  axes  with  the  origin  at  the 
centre,  to  which  when  the  hyperbola  is  referred,  its  eqiiation  will  take 
the  form  xy  =  c. 

Sug's. — The  common  equation,  A-y^  —  B^x'^  =  —  A^B^,  becomes,  when  we 
pass  to  general  oblique  axes  with  the  same  origin 


^22/2  4-  ^^  sin2  a    X2^  =  —  A'^B-^. 
—  B-  cos-  a 
B^  cos''^  a'  =  0,    and    A'^  sin^  a  — 

^  B 

Now,   in  order  that 


A^sin^a     yz'^ -\- 2 A- sin  a  sin  a 
—  B-  cos2  a'  —  2B^  cos  a  cos  a' 

The  conditions  imposed  are.   A-  sin^  a' 

B^  cos-  a  =  0  ;  whence  tan  a  =      ,    ,  and  tan  a'  =  — 7—. 

A  A 

these  values  should  indicate  the  positions  of  different  lines,  they  must  be  taken 

with  different  signs.     Thus  the  new  axes  are  found  to  make  angles  with  the  primi- 

tive  axis  of  x  whose  tangents  are  — — ,  and  — ,  which  relation  characterizes  asymp- 

totes.    The  equation  then  reduces  to 

(2  J.2  sin  a  sin  a'  —  2B^  cos  a  cos  a')x2y2  =  —  A'^B'^. 

But  the  conditions  above  give  sin  cc  =  — _.  cos  a 


:,  sm  oc 


B 


^A-2  +  B^ 


,  and  cosa' 


\/a^  +  B^ 


v/^2  +  52  \/^2  _|.  Bi 

These  values  substituted  in  the  last  equa- 


J.2  _L.  £2 

tion,  give,  after  reduction,  ^2^2  = 1 »  which  is  the  same  form  as  found 


before. 


90 


TRANSFORMATION  OF  CO-ORDINATES 


Ex.  15.  Letting  a;^  —  6xy-i-y^  —  6a:  +  2?/  +  5  =  0  repiesent  a  locus 
referred  to  rectangular  axes,  required  the  equation  when  the  refer- 
ence is  to  a  new  set  of  rectangular  axes  with  the  origin  at  (0,  —  1), 
and  the  new  axis  of  x  makes  an  angle  of  — 45°,  or  135°,  with  the 
primitive  axis  of  x. 

Svg's.— The  fonnulce  for  transformation  become,  in  this  case,  x  =  s/Hxz  +  2/2), 
and  y  =  \/iiy.2  —  Xz)  —  1.  The  transformed  equation  is  y^  —  2x2  =  2  (See 
Fig.  88,  and  the  illustration  accompanying  it.) 


^  »» 


SECTION  IL 

Methods  of  Passing  from  Eectilinear  to  Polar  Co-ordinates, 

and  vice  versa. 

12s,  IProh, — To  produce  the  formulce  for  passing  from  a  Rect- 
angular to  a  Polar  system  of  co-ordinates. 

Solution. — Let  P  be  any  point  in  a  locus 
M  N  referred  to  the  rectangular  axes 
A,Xi,  AiYj,  the  co-ordinates  of  P  being 
Ai  D  =x,  and  PD  =  y,  when  referred  to 
these  axes.  Let  the  pole  of  the  new,  or  polar 
system,  be  A  2,  whose  co-ordinates  are  m  and  n  ; 
and  let  A2X2,  or  AgXj',  be  the  polar  axis 
making  an  angle  a  with  AX,  or  what  is  the 
same  thing,  with  A2  K  parallel  to  AiXj. 
Let  the  polar  co-ordinates  of  P  be  A,  P  =  r, 


and   PA2X; 


PA2X2'  =  6.     The  angle 


Fig.  95. 


PA2  K  will  be  0  -j-  a:  when  the  polar  axis  lies  above  A2  K,  and  0  —  a,  when  the 

polar  axis    Hes    below ;    hence,    in    general,    PA2K    =   0    zb    a.      Now   x  = 

Ai  D  =  A,B  +  A2H.     But,  from  the  triangle  PAgH,  A2H  =  rcos  PA2  K  = 

r  cos  {Q  ±  a).     . ' . 

x^=m  -\-r  cos  (0  zt  a),  (1). 

In  like  manner  y  =  n  +  r  sin  (0  ±  a),  (2),  as  y  =  PO  =  AzB  -}- PV^, 

and  P  H  =  r  sin  (0  ±  a) . 

If  the  pole  is  at  the  primitive  origin,  m  =  0,  n  =  0,  and 

X  =  r  cos  (0  zh  a),  (1 1 )  ; 
and  y=zrsin{6±a),  (2i). 

If  the  polar  axis  is  parallel  to  the  primitive  axis  of  x,  a  =  0,  and  the  formulce 
become 

cc  =  m  -J-  rcos  0,   (Ig)  ; 

2/  =  n  -|-  r  sin  0,  (^2) ',  or,  if  the  pole  is  at  the  prim- 
itive origin,  x  =  r  cos  0,  (1 3)  ; 

2/  =  rsine,  (23). 


FEOM  RECTILINEAR  TO  POLAR  CO-ORDINATES,  AND  VICE  VERSA.      91 

12'9»  J[*Vob, — To  produce  the  formulae  for  passing  from  a  Polar  to 
a  Rectangular  system  of  co-ordinates. 


V— ' 


Solution.— From     Fig.   95     we    have     PA2    =    ^PH     +    A2H,     or 

r  =  s/ {y  —  rt)2  •\-  {X  —  m)2.     From  the  same  triangle  we  have  also  cos  {B  ±a)  = 

^  —  fn,  -,    .     ^         .  V  —  ■^  ,  .  , 

,  and  sm  (6  =h  a)  =  —  — 1,  which  are  the 


\/{y  —  n)2  -\-  {X—  my^  \/{y  —  n)^  +  (^c  —  m)2 

formulce  sought. 

When  the  polar  axis  is  parallel  to  the  primitive  axis  of  x,  the  formulce  are 


X  —  m 


r  =  \/(2/  —  ?i)2  -\-  (X  —  m)2,    COS0  =  —  — .    and   sin  0  = 

V{y  —  n)"^  +  (aJ  —  w)2 

HI  Yi 

If  at  the  same  time  the  origin  and  pole  coincide,  ih.e  formulas 


\\y  —  ny-\-{x  —  7)1)2 

are  r  =  \/y^  -f~  ^^j  ^^^  ^  =  — ^==z  >  and  sin  6 


s/yi  -^  cty^  s/y2  _|_  x2 

Ex.  1.  Transform  572 -f?/2= 5a j:  to  polar  co-ordinates,  the  pole  being 
at  the  origin,  and  the  polar  axis  coincident  with  the  axis  of  x. 

The  equation  is  r  =  5a  cos  0. 

SuG. — The  formulce  are  x  =  r  cos  0,  and  y  =  r  sin  0. 

Ex.  2.  Transform  {x^  +  y-)'^  =  a'^{x^  —  y"^)  to  polar  co-ordinates, 
the  pole  being  at  the  origin,  and  the  polar  axis  coincident  with  the 
axis  of  X. 

The  equation  is  r~  =  a^^cos^  0  —  sin^  6)  =  a'  cos  2d. 

Ex.  3.  Transform  r^  =  a^cos20  to  {x^  +  y^y  =  a^{x^  —  y^).  (See 
last  example.) 

SuG. — First  put  the  equation  in  the  form  r^  =  a^{cos^  0  —  sin^  0). 

Ex.  4.  Under  the  same  conditions  as  above  transform  r^  cos  26  =  a'-* 
to  072  —  2/3  =  aK     Also  xy  =  a^,  to  r^  sin  20  =  2a^.     Also  ^2  _{_  ^/a  = 

(2a  —  a;)  2,  to  r^  cos-|(?  ■=  a^.     Also  reverse  these  processes. 

Ex.  5.  To  deduce  from  A^y^  +  B^x^  =  A-^B^,  the  polar  equation  of 
the  ellipse,  in  terms  of  the  transverse  axis  and  eccentricity,  the  pole 
being  at  the  left  hand  focus  and  the  polar  axis  falhng  on  the  axis  of 
the  curve. 

Sug's. — The  given  equation  being  put  in  the  form  y^  =  (1  —  e^){A^  —  x^),  and 

the  formulce  for  transformation  in  the  form  x  =  r  cos  0  —  Ae,  and  y  =  r  sin  0,  an^ 

the  substitutions  made,  we  have 

r2sin2  0  =  (1  —  e2)(^9  —  r2cos2  0  -\-  2J.ercos0  —  AH^). 

Expanding  this  and  reducing  to  a  known  form, 

„      2Jecos  0(1  —  e2)        A^l  —  e^y    , 

r^ _ : — r  =. ;  hence 

1  —  e2 cos2  0  1  —  6200820 


92  TRANSrOllMATlON    Or   CO-Or.DINATES. 


_  AecosBjl  —  e2)  rh  V A'^jl  —  e^)-^(l  —  e^cos-e>  -j-  A^e^cos-Q^l  —  e'-J)^ 

1  —  e'^  cos-'  6 


^e  cos  6(1  —  e2)  d=  v/^-^(l  —  e^)^  _  ^e  cos  9(1  —  e^)  ±  ^(1  —  e^)  _ 
1  —  e^  cos'-'  0  1  —  e2cos'-^6  ~ 

— ^^ — -.     Now  as  neither  e  nor  cos  9  can  exceed  1,  and  as  each  is 

1  —  e-  cos-^  6 

generally  less  than  1,  r  is  positive  only  for  e  cos  6  + 1  5  hence  we  may  reject  the  — 

sign  m  this  factor  and  write  r  = ; = ... 

°  1  —  e-  cos-2  6  1  —  e  cos  6 

^(1 e2) 

If  the  pole  is  taken  at  the  right  hand  focus,  x  =  rcosQ  +  Ae,  and  r  =  --^, — . 

^  l-j-ecos0 

(See  107 -109.) 

[Note. — There  are  expedients  by  which  the  algebraic  reductions  in  this  solution  may  be  simph- 
fied  ;  but  as  our  purpose  is  to  exhibit  simply  the  process  of  transformation,  we  do  not  think  best 
to  avoid  the  work  by  indirect  means.  Were  the  object  merely  to  obtain  the  polar  equation,  the 
process  of  {107 — 109)  would  be  much  more  simple  and  elegant.] 

P 

Ex.  6.  Deduce  the  polar  equation  of  the  parabola,  r  = -, 

^  1  —  cos  0 

from  2/2  =  2px.     (See  108.) 

Ex.  7.  Transform  the  equation  of  the  cissoid,  y^  = ,  to  the 

2a(l — cos2<?)  r»     •    ^i      ^     /o      -.^^  N 

polar  equation,  r  =  — ^ ,  or  r  =  2a  sm  6  tan  6.    (See  110,) 

^  ^  cos^' 


GEIVERAL  SCHOLIUM. 

The  student  being  now  famiUar  with  equations  as  representatives  of  loci, 
is  prepared  to  use  them  as  instruments  for  the  investigation  of  the  properties 
of  their  loci.  But,  in  carrying  forward  this  work,  the  Calculus  renders 
great  assistance,  and  for  many  purposes  is  indispensable.  Therefore  before 
commencing  the  next  chapter,  the  student  must  become  familiar  with  the 
processes  of  differentiating  the  various  kinds  of  explicit  and  implicit  func- 
tions of  a  single  variable,  with  successive  differentiation,  partial  differentia- 
tion, the  development  of  functions  by  Maclaurin's  and  Taylor's  theorems, 
the  evaluation  of  indeterminate  forms,  and  the  theory  of  Maxima  and 
Minima,  as  treated  in  the  first  two  chapters  of  the  second  part  of  this  vol- 
ume. Having  read  those  chapters,  he  can  turn  back  and  resume  his  study 
of  Geometry  at  this  point.  Students  who  do  not  choose  to  study  the  Cal- 
culus, may  complete  their  course  in  this  subject  by  reading  Sections  XlVr 
and  XY.  of  the  next  chapter  in  this  part. 


CHAPTER  IV. 

JPItOrERTIES    OF   PLANE   LOCI   INVESTIGATED   BY 
MEANS  OF  THE  EQUATIONS   OF  THOSE  LOCI. 


SUOTIO]V  I. 
Tangents  to  Plane  Loci. 

(a)     BY  IIECTILINEA.K   CO-OEDINATES.  ' 

130,  Bef. — Consecutive  J^oints  on  a  line  are  points  nearer 
to  each  other  than  any  assignable  distance. 

IiiL.  — As  we  shall  have  frequent  occasion  to  use  this  conception,  it  is  of  the 
utmost  importance  that  the  definition  be  clearly  comprehended.  For  example, 
when  we  speak  of  P  and  P',  Fig.  96,  as  consecutive  points,  we  do  not  conceive 
them  as  absolutely  in  juxtaposition,  i.  e.,  so  near  each  other  that  there  can  be  no 
other  point  between  them  ;  but  we  mean  simply  that  we  are  to  reason  upon  them  as 
nearer  each  other  than  any  assignable  distance.  In  short,  PP'  is  merely  to  be 
considered  infinitesimal  in  the  sense  of  being  less  than  any  assignable  distance. 
.So,  also,  when  we  speak  of  PD  and  P'D'as  consecutive  ordinates,  we  mean  that 
D  D'  is  to  he  treated  in  the  argument  as  less  than  any  assignable  distance — as  infin- 
itesimal. 

131,  Def. — A.  Tangent  (rectilinear)  is  a  right  line  passing 
through  two  consecutive  points  of  a  curve. 

ScH.  — For  many  purposes,  it  is  more  convenient  to  speak  of  the  two  con- 
secutive points  through  which  a  tangent  passes,  as  one  point,  and  call  it  the 
point  of  tangency :  this,  of  course,  is  necessarily  the  case  when  the  expression 
is  in  finite  terms,  as  the  distinguishing  of  consecutive  points  requires  infin- 
itesimals. The  student  has  become  familiar,  in  Elementary  Geometry, 
with  the  conception  of  a  curve  as  a  polygon  of  an  infinite  number  of  infin- 
itesimal sides  :  the  prolongation  of  one  of  these  sides  may  be  considered  a 
tangent. 

132,  CoR. — A   Tangent  has  the  same  direction  as  the  curve  at  the 

point  of  tangency. 


133,  JProp. — The  first  differential  coefficient  of  the  ordinate  of  a 
curve  regarded  as  a  function  of  the  abscissa  (^)  is  the  tangent  of  the 
angle  which  a  tangent  to  the  curve  makes  with  the  axis  of  abscissas. 


94 


PROPERTIES  OF  PLANE  LOCL 


Dem.— Let  MN,  Fig.  96,  be  any  plane 
curve  whose  equation  is  y  =^f{x).  Let  P  and 
P'  be  consecutive  points,  and  PD  and  P'  D' 
consecutive  ordinates.  Then  is  RS,  drawn 
through  P  and  P',  a  tangent.  Draw  PE 
parallel  to  X'X.  Since  P  and  P'  are  con- 
secutive points,  DD'  and  P'E  are  contem- 
poraneous infinitesimal  increments  of  x  and 
y,  respectively;  i.  e.,  DD'  represents  dx, 
and  P'E  represents  dy.  Now,  letting  STX  be  represented  by  a,  we  have 
P'E       dy 


Fig.  96. 


tan  a  =  tan  P'  P  E  = 


PE 


dx' 


Q.   E.   D. 


Ex.  1.  What  angle  does  a  tangent  to  the  curve  y  =  x^  —  ^2  + 1,  at 
the  point  a:  =  2,  make  with  the  axis  of  ^  ? 

Solution.     -^  =  dx-^  —  2x,  which,  for  a  =  2  becomes 

dx 

-^  =  8,  using  (cc'j  y')  to  designate  the  particular  point 

CLtC 

in  distinction  from  the  general  point  {x,  y).  .  • .  The 
tangent  at  the  point  x  =  2,  makes  an  angle  with  the  axis 
of  X  whose  tangent  is  8,  i.  e.,  an  angle  of  82°  52'  30". 
The  figure  is  that  in  the  margin,  in  which  P  is  the 
point  of  tangency,  and  PXX  is  the  angle  whose  tan- 
gent is  8.     (Tan-i8  =  PTX.) 

Ex.  2.  At  what  point  does  the  curve  in  the 
last  example  run  in  a  direction  making  an  angle 
of  45°  with  the  axis  of  iP?     At  what  point  does 
it  make  an  angle  of  135°?     At  what  point  is  it  perpendicular?     At 
what  point  parallel  ? 

SuG. — The  direction  of  the  tangent  being  the  same  as  that  of  the  curve,  we  have 

dy 
simply  to  find  where  — ,  or  3x2  —  2x  (which  is  its  general  value  in  this  curve) 

equals  1,  0,  — 1,  or  00,  as  these  are  the  tangents  of  the  required  angles.  Thus  for 
the  first  we  have  to  find  the  value  of  x  which  satisfies  Sx^  —  2a;  :=  1.  This  gives 
£C  =  1  and  —  i.  Now,  for  a;  =  1,  ?/  =  1  as  is  found  by  substituting  in  the  equa- 
tion of  the  curve.  This  point  is  P'  in  the  figure.  The  curve  also  runs  in  the 
same  direction  at  (  —  3,  f?^),  P"  in  the  figure. 


Fig.  97. 


Answers,  The  curve  is  parallel  at  (0,  1),  and  (f,  f^) 

i±n/: 


It  makes  an 


angle  of  135°  at  x 


3 


,  which  being  an  imaginary  point 


signifies  that  the  curve  in  this  plane  does  not  make  an  angle  of 
135°  with  the  axis  of  x.  It  is  perpendicular  at  ^  =  +  00,  and 
—  00  ;  i.  e.,  as  the  curve  extends  to  the  right  or  left  it  approaches 


TANGENTS — BY   rtECTILINEAK   CO-ORDINATES.  93 

perpendicularity  with  the  axis  of  x,  but  becomes  pei-pendicular 
only  at  an  iniinite  distance  from  the  origin. 

134,  ScH. — To  determine  at  what  point  on  a  given  curve  a  tangent  must 

be  drawn  to  make  a  given  angle  with  the  axis  of  x,  — or,  what  is  the  same 

thing,  to  find  a  point  at  which  a  curve  has  a  given  direction, — put  the  value 

di/ 
of  -^  as  derived  from  the  equation  of  the  curve,  equal  to  the  tangent  of 
ax 

the  given  angle  or  direction,  and  solve  this  equation  in  connection  with  the 

equation  of  the  locus.     To  find  where  the  curve  is  parallel  to  the  axis  of  x, 

dy 
put  the  value  of  -^  equal  to  0,  and  solve  as  before.     To  find  where  it  is 
dx 

dy 
perpendicular,  put  the  value  of  —  equal  to  co,  and  solve  in  the  same  way. 

dx 

Ex.  3.  At  what  point  on  the  curve  y-  =  2x^,  does  a  tangent  make 
an  angle  with  the  axis  of  x,  whose  tangent  is  3  ?  At  what  point  is 
the  curve  parallel  to  the  axis  oi  x?  Where  is  it  perpendicular? 
What  is  the  direction  of  the  curve  at  j;  =  8  ?  Construct  the  figure 
and  observe  the  agreement  of  results  therewith. 

Answers,  At  (2,  4);  at  (0,  0);  nowhere;  tan~^(  ±  6).  The  last 
result  indicates  two  tangents  corresponding  to  ^  ==  8,  one  drawn 
through  (8,  32),  and  the  other  through  (8,  — 32). 

Ex.  4.  At  what  point  in  the  curve  y'^  =  2x  -{■  Sx^  must  a  tangent  be 

drawn  to  make  with  the  axis  of  x  an  angle  whose  tangent  is  ^  ?  1  ?  2  ? 

Answers.  The  first  two  are  impossible.     How  does  this  appear  in 

the  solution,  and  how  from  the  locus  ?     The  tangent  of  the  angle 

is  2  at  (^,  1),  and  at  (—1,  —1). 

Ex.  5.  At  what  point  on  y  ==  x^  —  3x^  —  24a;  +  85  is  the  tangent 
parallel  to  the  axis  of  abscissas  ? 

Am.,  At  (4,  5),  and  at  (—2,  113). 

2 

Ex.  6.  Eind  in  the  curve  y  =■  a  -\-  2{x  —  b)^,  the  point  at  which  a 
tangent  is  perpendicular  to  the  axis  of  x.  Result,  At  (6,  a). 

X 

Ex.  7.  Under  what  angle  does  y  = cut  the  axis  of  abscissas  ? 

Stjg's. — As  tliis  curve  cuts  the  axis  of  a;  at  (0,  0),  tlie  question  is,  What  is  its 

direction  at  that  point?      Now  -^  = '■ — ,  which,   for  x  =  0,  becomes  1. 

.  • .  The  curve  cuts  the  axis  of  x  at  an  angle  of  45°. 

Ex.  8.  Show  that  the  sinusoid  cuts  the  axis  of  x  alternately  at  45° 
and  135°.  -' 

Ex.  9.  What  angle  does  the  focal  tangent  of  the  common  parabola 
make  with  the  axis  oi  x'> 


96 


PROPEETIES  OF  PLANE  LOCI. 


13S,  Cor.— ^  the  axes  are  oblique,  -^  signifies  the  ratio  of  the  sine 
of  the  angle  which  the  tangent  makes  with  the  axis  of  x,  to  the  sine  of  the 


angle  which  it  makes  with  the  axis  of  j,  i.  e., 


sin  a 


sm  (/j  —  a)' 


(See  34.) 


136,  I^f'Op.—Tne  general  equation  of  a  tangent  to  a?iy  plane  curve  is 

f  in  which  (x',  y')  is  the  point  of  tangency,  and  x  and  j  are  the  current 
co-ordinates  of  the  tangent. 

Dem.— Let  MN,  Fig.  98,  be  any  plane 
curve  whose  equation  is  y  =f(x),  and  let  P 
be  the  point  of  tangency  whose  co-ordinates 
are  x',  y'.  Now,  the  equation  of  any  line 
passing  through  {^x  ,  y')  is  y  —  y'  =  a{x  —  x') 
(32).  But,  in  order  that  this  line  should  be 
tangent  to  M  N  at  P,  the  tangent  of  the  YL^T 
angle   PTD,  which  is  represented  by  a  in      " 

the  formula,  must  be  -^.      Hence,  substitu- 
ax 


Fig.  98. 


ting,  we  have  y  —  y'  =■  ^{x 


X'). 


l.  E.  D. 


Ex.  1.  "What  is  tlie  equation  of  a  tangent  to  an  ellipse  referred  to 
its  axes? 

Solution. — The  equation  of  the  locus  is  A^y-  -\-  B^x^  =  A^B'^ ;  whence  y^  =  — 

B-x                                                                       B'^x'  dv 

—r—,  which  satisfied  for  the  point  (x\  y')  is -—.     Substituting  this  value  of  — 


in  the  general  equation  of  a  tangent  (130),  we  have  y  —  y'  = 


jB%' 


A^y' 


X  —  .T 


Keducing,   this  becomes  A^yy' -{- B^x'  =  A^y' ^  +  B^x'-^.      But  as  (x,  y')  is  a 
point  in  the  locus  A^'^  -{-  B^x''^  =  A^B'^  ;  hence,  finally,  A-t/t/'  -|-  B'^hcx'  =  A^B'^. 


dx 


_  ^     dy  B^x 

QuEEiEs. — Li  —  = —  , 

dx  A^y 


,  what  are  x  and  y  co-ordinates  of?    In.  y  —  y'  = 


:{x  —  x'),  what  are  x  and  y  co-ordinates  of?     "What  x'  y'  ?     Of  what  degree  with 


respect  to  the  variables  is  A-yy'  -f-  B-xx  =  A-B-?  Why  should  it  be  of  this 
degi-ee  (5«5)  ?  What  are  the  variables?  Notice  that  fo7^  the  same  tangent,  x  and  y 
have  all  values,  but  x'  and  y'  have  fixed  values  :  x',  y'  are  general,  i.  e. ,  they  rep- 
resent any  point  in  the  locus,  but  they  do  not  represent  all  points  at  the  frame 


x'),  we  had  chanced  to  find 


time.     If  in  our  deductions  from  y  —  v'  =  -r-.i'X' 

^        dx 

A^^  -f-  B-x'^  could  we  have  substituted  for  it  A^B%  as  we  did  for  A-y''^  -f-  B-x'-  ? 

Why  not? 


TANGENTS- 


KECTILINEAR    CO-ORDINATES. 


97 


Ex.  2.  Produce  the  equation  of  a  tangent  to  dy^  -{-x^  =  5,  at  a;  =  1, 
and  construct,  first,  the  tangent  from  its  equation,  and,  second,  the 
curve  from  its  equation. 


Solution. 


-In  this  locus  -^  = 
ax 


< —  — .     This  is  the  general  value 
32/ 

of   the    tangent    of   the    angle 

■which  a  tangent  to  this  ellipse 

makes  with  the  axis  of  x.     For 

the  particular  point  x  =  1  (for 

which  y  =  ±z  1.155,  nearly),  we 

have  -p,  =  =F  .29  approximately. 


Fig.  99. 


Substituting  these  values  in  the  general  equation  of  a  tangent  {136),  we  have 
y  =F  1.155  =  zp  .2d[x  —  1),  or  y  =  =p  .2d.x  ±  l.M.  There  are,  therefore,  two 
tangents  to  this  locus  at  a;  =  1  ;  one  whose  equation  is  ?/  =  —  .2dx  -f-  1.44,  and 
another  whose  equation  is  y  =  .29x —  1.44.  RS,  in  the  figure,  represents  the 
former;  and  R'S',  the  latter. 

Another  solution  of  this  example  is  obtained  by  observing  that  the  locus 
Sy-  -|-  35^  ==  5  is  an  ellipse  whose  semi-axes  are  A  =  s/d,  and  B  =  v/f.  But  the  equa- 
tion of  a  tangent  to  an  ellipse  is  A^py'  -\-  B'xx  =  A^B^-  ;  whence,  substituting, 
we  have  hyy'  -j-  ^xx  =  ^3^,  or  Zyy'  -{-  xx  =  5,  as  the  equation  of  any  tangent  to 
this  eUipse.  For  the  points  (1,  ±  1.155)  this  becomes  yz=^  .29a;  dr  1.44,  as 
before. 


Ex.  3.  Deduce  the  equation  of  a  tangent  to  an  hyperbola.     Also 
of  a  circle.     Also  of  a  parabola. 

r  The  equation  of  a  tangent  to  an  hyperbola  is 

1       "         "  "         "         "  a  circle  is  yy'  +  xx'  =  R\ 

^      "         "  "         "         "  a  parabola  is  y?/'  =  p(^+ ^')- 


Besults. 


Ex.  4.  "What  is  the  equation  of  a  tangent  to  the  parabola  y^  =  dx 
at  .r  =  4  ?  Construct  the  tangent  from  its  equation,  and  then  con- 
struct the  parabola  as  in  Ux.  2. 

For  (4,  6)  the  equation  is  y  =  f  ^  +  3. 

Is  there  another  tangent  for  a;  ==  4  ? 

Ex.  5.  Produce  the  equation  of  a  tangent  to  3y^  —  2x^  =  10,  at 
^  =  4.  Is  there  more  than  one  tangent  ?  Construct  the  figure  as 
above.  Equation,  y  =  dz  .1121  x  ±  .8909. 

Ex.  6.  What  is  the  equation  of  a  tangent  to  t/^  ==  4  —  .r',  at  a:  =  3  ? 
"Why  is  the  result  imaginary  ? 


98  PKOPEKTLES  Uv    PLANE  LOCI. 

Ex.  7.  What  is  the  equation  of  a  tangent  to  y^  = ,  at  a:  =  2  ? 

Ans.,  y  ==  2x  —  2,  and  y  ==  —  2:r  +  2. 

Ex.  8.  Show  that  the  equation  of  the  tangent  to  the  Napierian 
logarithmic  curve  {x  =  log  y)  is  y  =  y\x  —  x'  -f  1).  Observe  that 
the  ordinate  to  this  curve  at  any  point,  is  the  natural  tangent  of  the 
angle  which  the  tangent  to  the  curve  at  that  point  makes  with  the 
axis  of  abscissas. 

Ex.  9.  What  is  the  equation  of  a  tangent  to  an  hyperbola  referred 

y< 

to  its  asymptotes  {xy  =  m)  ?  Ans.^  V  = ,^  +  2?/'. 

y' 

Interpretation  of  the  equatioji  y  = -,x  -j-  2^/'.      If  this  represents  the  tangent 

y' 

to  an  hyperbola  referred  to  rectangular  asymptotes, r,  is  the  tangent  of  the 

angle  which  the  tangent  to  the  curve  makes  with  the  axis  of  x  ;  and  2y  is  the  dis- 
tance from  the  origin  at  which  the  tangent  cuts  the  axis  of  y,  as  in  all  equations 
of  right  hues  referred  to  rectangular  axes.  But  in  this  case  the  hyperbola  is  equi- 
lateral, since  xy  =  m  is  the  equation  of  an  equilateral  hyperbola  when  the  asymp- 
totes are  rectangular  ;  or,  in  other  words,  no  hyperbola  but  an  equilateral  one  has 
rectangular  asymptotes To  interpret 

v' 
2/  =  —  —x  4-  2^/'  for   oblique    axes,  we 

CI/ 

observe  that  2y'  is  AO,  Fig.  100;  and  by 
making  ?/  =  0  we  find  that  the  intercept 
on  the  axis  of  x,  AT",  is  2x' .     Now  the 

coefi&cient  of  x,  —  - ,  is  the  ratio  of  the 

X' 

sine  of  the  angle  which  the  line  (tangent) 

makes  with  the   axis  of  x   to  the   sine 

of  the  angle  which  it  makes  with  the 

axis  of  y,  by  {34:).      This  fact  accords  Fig.  100. 

with  the  relations  observable  from   the 

,      .  ^-^    AO       sin  A  TO  y'        sin  A  TO 

figure.     Thus,  m  the  tnangle  AOT,   ^T^sinAOT'  °'    ^  =  sin  AOT* 

The  minus  sign  is  explained  by  observing  that  the  line  R  S  lies  across  the  1st 
angle  when  P  is  in  this  angle,  and  to  pass  to  this  position  from  that  in  the  funda- 
mental figure.  Fig.  24,  the  angle  NGX  of  that  figure  becomes  STX  of  this,  an 
angle  whose  sine  is  +,  and  equal  to  sin  ST  A.  But,  by  this  change  of  the 
position  of  the  line,  the  angle  G  H  A  of  Fig.  2-i,  first  diminishes  to  0  and  then 
re-appears  generated  from  left  to  right  and  henc-^  is  a  negative  angle.     Therefore 

-f-  sin  ATO  y' 

sin  A O T  is  negative,  and  we  have : — .  ^_,  = ;. 

°  —  sm  AOT  X 

Ex.  10.  Produce  the  equation  of  a  tangent  to  the  locus  y^  =  2x  -]r 
Zx^,  at  ^  =  2. 

Result,  There  are  t^o  tangents,  viz.  :  i/  =  ±  |j7  ±  -J^ 


TANGENTS — BY   BECTILINEAll   CO-OEDINATES.  99 

Ex.  11.  At  "wliat  angle   does  the   line  y  n=  i.^;  -f-  1  cut  the  curve 
r/2  =  4a7?  Ans.,  10°  14',  and  33°  4'. 

Sug's. — Find  the  point  of  intersection  and  the  tangent  to  the  curve   at  this 
point.     Find  the  angle  included  between  this  tangent  and  y  ■=  hx  -\-  Ihj  (36). 

Ex.  12.  At  what  angle  does  y-  =  lOo;  intersect  x^  -]-  y^  :=  144  ? 

Ans.,  71°  0'  58". 

Ex.  13.  At  what  angle  does  25?/2  -f  16^2  =  1600  intersect  IGif  — 
9x^  =  _  576  ?  Ans,    61°  58'  37". 


13  7 •  JPfop, — The  general  value  of  the  intercepts^  of  the  axes  by  a 
tangent  are 

.dx' 


'dy' 


^  —  y\j.,,^ 


and  Y=y'  —  x'£^ ; 

in  which  X  is  the  intercept  on  the  axis  of  x,  and  Y  that  on  the  axis  of  y, 
(x'j  y')  being  the  point  of  tangency. 

Dem.— The  equation  of  a  tangent  being  y  —  y'  =  -r-:(x  —  a;'),  if  we  find  where 

(JLvu 

this  line  cuts  the  axes  by  making  2/  =  0  for  the  intercept  on  the  axis  of  x,  and  finding 

the  value  of  x ;  and  cc  =  0  and  finding  the  value  of  y  for  the  intercept  on  the  axis  of 

y,  we  have  the  results  sought.     Thus  for  y  =  0,  and  a:  =  X,  we  have  0  —  y'  = 

dv '  dx' 

-7^(X-=-  a;'),  or  X  =  x'  —  v'-r-.     For  x  =  0,  and  y  =  Y.  we  have  Y —  y'  = 

dx  dy 

-^,(0  —  X'),  or  F=  y'  —  a'^.     Q.  e.  d. 
dic  ^  '  -^  dx' 

ScH. — In  solving  an  example  we  may  either  apply  these  formulae, ;  or, 
first  get  the  equation  of  the  tangent  and  then  make  x  and  y  successively 
=  0.     This  is  but  an  application  of  [26.,  1st). 

Ex.  1.  From  A^y"^  +  B-x"-  =  A^B"^,  show  that  a  tangent  to  an  ellipse 

A^  B-i 

cuts  the  axes  at  X  -==  — -,  and  F  =  — ^ ;  i.  e.,  If  from  any  point  in  an 

ellipse  a  tangent  and  an  ordinate  be  drawn  to  either  axis,  half  that  axis  is 
a  mean  proportional  between  the  distances  of  the  intersections  from  the 
centre. 

*  This  is  an  abbreviated  form  of  expression  for  "the  distances  from  the  origin  to  where  the 
curve  cuts  the  axis." 


100 


Hi-. 

PEOPERTIES   OF   PLANE   LOCI. 


Sug's.— In  the  figure,  AT  =  X,  AD  =x', 
PD=y  ,  AO  =  Y,  AB=A  and  AG  =  jB. 

Hence  having  obtained  X  =  — ,  we  have  but 

to  put  it  into  the  form  X  :  ^  ::  ^  :  a;',  to  ob- 
serve   the    truth   of   the    proposition.      Also 


52 
r  =  -7  gives  Y  :  B 

y 


JB  :y' 


138,  ScH.- 


A^ 


-Since  ^  =  — .  we  see  that 

X 


Fig.  101. 


the  intercept  of  the  axis  of  x  does  not  depend  upon  the  conjugate  axis  of 
the  ellipse,  so  that,  if  on  the  same  transverse  axis,  different  ellipses  be  drawn, 
the  intercejjts  on  this  axis,  hy  tangents  corresponding  to  the  same  abscissa  are 
equal.  That  is,  if  x'  and  A  remain  the  same,  AT  is  the  same.  From  this 
"we  have  a  ready  method  of  drawing  a  tangent  to  an  ellipse  geometrically. 
Thus,  let  it  be  required  to  draw  a  tangent  to  the  ellipse  Fig.  101,  at  the 
point  P.  Draw  a  circle  (a  variety  of  ellipse)  upon  the  same  transverse 
axis.  Draw  the  ordinate  PD  and  produce  it  to  P'.  Draw  a  tangent  to  the 
circle  at  P'.  This  fixes  the  intercept  AT.  Draw  a  line  through  P  and  T 
and  it  is  the  tangent  sought. 

[Note — The  student  should  make  himself  perfectly  familiar  with  this,  and  all  methods  given  for 
drawing  tangents  to  loci  geometrically.] 

Ex.  2.   Show  that  in  the  hyperbola  the  intercepts  on  the  axes  made 

A"  B^  ... 

by  a  tangent  are  X  =  — -,  and  Y= j,  and  that  the  proposition  in 

X  y 

Ex.  1,  is  true  also  of  the  hyperbola. 

139,  ScH. — This  principle  also  affords 
a  method  of  drawing  a  tangent  to  an  hy- 
perbola geometrically.  From  the  given 
point  of  tangency  P,  let  fall  the  ordinate 
PD  ;  and  upon  the  transverse  axis  HB, 
and  the  abscissa  AD,  draw  semi-circum- 
ferences. From  their  intersection  let 
fall  LT  a  perpendicular  upon  the  axis 
of  X.  Draw  a  line  through  P  and  T  and 
it  is  tangent  to  the  curve  at  P.     Proof.  Drawing  AL  and  LD,  we  have 


Fig.  102. 


AD  (or  x)  :  AL  (or  J)  :  :  AL  (or  A)  :  AT. 
intercept  made  by  a  tangent  at  P. 


Whence  AT  ==  —   and  is  the 

x' 


Ex.  3.  Prove  that,  if  a  tangent  be  drav^n  to  a  parabola  at  any  point, 
the  intercept  on  the  axis  of  x  is  equal  to  the  abscissa  of  the  point  of 
tangency. 


8UBTANGENTS — BY  BECTILINEAR  CO-OEDINATES. 


101 


14:0 •  ScH. — The  principle  developed  iu  the  solu- 
tion of  this  example  affords  the  most  simple  method 
of  drawing  a  tangent  to  the  parabola,  geometrically. 
Let  it  be  required  to  draw  a  tangent  at  P,  Fig.  103. 
Draw  the  ordinate  PD,  take  AT  =  AD,  and 
through  P  and  T  draw  a  line.  This  will  be  the 
tangent  required. 


Ex.  4.  To  find  where  the  tangent  to  y^x  = 
4(2  —  x)  (the  witch  of  Agnesi,  the  radius  of 
the  fixed  circle  being  1),  at  ^=  2,  cuts  the  axes. 

Besults,  It  cuts  the  axis  of  ^  at  a;  =  2,  and  the  axis  of  y  Sit  y 
i.  e.,  is  parallel  to  the  latter. 

[Note. — Observe  from  the  last  example  that  a  tangent  may  cut  the  curve.] 


Fig  103. 


00, 


141,  Def. — Tlie  Subfangent  is  the  portion  of  the  axis  of 
abscissas  intercepted  between  the  foot  of  the  ordinate  from  the  point 
of  tangency,  and  the  intersection  of  the  tangent  with  this  axis  ;  or  it 
may  be  defined  as  the  projection  of  the  corresponding  portion  of  the 
tangent  upon  the  axis  of  abscissas.  In  each  of  the  three  preceding 
figures  DT  is  the  subtangent  corresponding  to  P  as  the  point  of 
tangency. 


14:2,  I^TOp, — The  general  value  of  a  subtangent  is 

dx' 

Suht  =  v'-T", 

^  dy' 

in  which  (x',  y')  is  the  point  of  tangency. 


Dem.— In  auy  of  the  three  preceding  figures  we  have  from  the  triangle    PTD, 

DX  =  PD  X  cot  PXD.      But  DT"  is  subt,  PD  =  y',  and,  as  tan  PTD  is 

^y'        .  ^-^^  .    dx'  ,,         dx' 

-/-,,  cot  PXD  is  -;— ,.      .-.  suht  =  2/  -r— .     Q.  E.  D. 

dx  dy  dy' 

Ex.  1.  What  is  the  value  of  the  subtangent  of  y^  =  3a;2  —  12,  at 
a7  =  4? 

SuG  s. — For  a;  =  4,  V  =  ±  6.     —■  =  —•     .  * .  2/=  ±  6,  r—  =  -tx-  =dtzi;  and 
^  dx       y  dy         12 

dx' 
Suhi.  =^y'  -^  =  3.     The  pupil  should  construct  the  figure,  if  he  is  not  sure  that 

he  fully  comprehends  the  example  without. 

Ex.  2.  Find  the  value  of  the  subtangent  of  the  common  parabola. 
Of  the  logarithmic  curve.  Results,  2x\  and  m  or  1. 


102 


PROPERTIES   OF   PLANE  LOCI. 


Ex.  3.  What  is  the  value  of  the  subtangent  of  3/2  =  2^'  at  a:  =  2  ? 


5  S' 


Ex.  4.  If  upon  the  same  transverse  axis  different  ellipses  be  drawn, 
prove  that  the  corresponding  subtangents  are  equal. 


Sug's.  — The  general  value  is  SuU.  ■■ 
A^B^  —  B^x'-^       A^  —  £C'2 


■  B'X' 


,  ,      ,  a  result  which 

B^x  X 

does  not  depend  upon  B.      This  truth  is  q 

illustrated    in   Fig.   104,    DT"   being    the 

common   subtangent  for  all  the   ellipses, 

corresponding   to   the    same   value  of   a:, 

AD.     This  is  essentially  the  same  truth 

as  was  brought  to  light  in  Ex.   1,   Art. 

137. 


Fig.  104. 


Ex.  5.  Find  the  subtangent  to  the  hyperbola  referred  to  oblique 
asymptotes.  Result,  Subt  =  x'. 

ScH. — In  Fig.  100,  P  being  the  point  of  tangency  {x,  y'),  DT  =  Subt.  = 
X  =  AD.  Now  since  PD  is  parallel  to  AO,  PT  =  OP  ;  i.  e.,  The  inter- 
cepts of  a  tangent  to  a  hyperbola  between  the  point  of  tangency  and  the 
asymptotes  are  equal.  This  affords  a  method  of  drawing  a  tangent  when  the 
asymptotes  are  given.  Thus  let  it  be  required  to  draw  a  tangent  at  P, 
Fig.  100.  Draw  PE  parallel  k)  AX,  and  take  AT  =  2PE.  Through 
P  and  T  draw  a  Une  and  it  is  the  tangent  required. 


14:3,  JPvop, — The  general  expression  for  the  length  of  a  tangent, 
i.  e.,  for  the  portion  intercepted  between  the  point  of  tangency  and  the 
axis  of  X,  is 

Tan  =  y'Ax  +  — , 


in  which  (x',  y')  is  the  point  of  tangency. 


dy' 


Dem. — In  any  one  of  Figs.  101,  102,  103,  we  have  from  the  right  angled  triangle 


PDT,  PT   =  PD    +  DT  ,  or  PT 

dx' 


PD    +  DT 


PD=2/',and  DT  =  2/' 


dy'' 


rr.  I  dX^  ,  I  ,        dx"- 


Now  PT  is  Tan., 
dx"- 

—  -.       Q.   E.  D. 


Ex.  1.  What  is  the  length  of  the  tangent  to  1/2  =  2a:  at  ;r  =  8  ? 

Ans.,  4:VV7, 


ASYMPTOTES — BY  RECTILINEAR   CO-ORDINATES. 
Ex.  2.  Show  that 
In   the    eUipse,    Tan  =  ^^'^^pZI^  . 


103 


In  the  hyperbola,  Tan  =  y 


Ap^  -f  py^  -\-  Ay^ 


Ap'^  +  py^ 

In  the  parabola,  Tan  =  -vp'^  +  2/" ;    p   representing    the    semi- 

P 
parameter  in  each  case. 


14:4:,  Def. — A.n  ALsyaiptote  is  a  line  toward  which  a  curve 
constantly  approaches,  but  under  such  a  law  that  they  will  never 
meet ;  or^  what  is  the  same  thing,  that  they  will  meet  only  at  an 
infinite  distance  from  the  origin. 

An  asymptote  is  also  conceived  as  a  tangent  to  a  curve  at  an  infinite 
distance  from  the  origin,  which  yet  passes  within  a  finite  distance, 
i.  €.,  cuts  one  or  both  axes  making  finite  intercepts. 

Iiiii. — It  is  quite  common  for  persons  encounter- 
ing this  idea  for  the  first  time,  to  repudiate  it  as 
an  absurdity  ;  but  the  following  illustrations  will 
familiarize  it.  Let  the  law  of  the  curve  be  such 
that,  if  ordinates  Bb,  Cc,  Dd,  Ee,  Ff,  Gg,  etc., 
be  drawn  at  equal  distances  from  each  other,  each 
succeeding  ordinate  shall  be  a  the  preceding.  It  is 
evident  that  the  curve  will  continually  approach 
AX  but  under  such  a  law  that  it  can  never 
absolutely  reach  it.  {Practically  such  a  curve  will 
Boon  become  indistinguishable  from  the  line,  that 
is,  will  run  into  it.)  AX  is  an  asymptote  to  this 
curve. 

In  a  similar  manner  two  curves  may  approach  each  other  under  such  a  law  that 
the  distance  between  them  shall  constantly  diminish,  and  yet  the  curves  never 
meet.  Such  curves  are  asymptotes  to  each  other.  Our  present  purpose  embraces 
only  rectiUnear  asymptotes. 


B    C    D   E    F    G 

Fig.  105. 


H     X 


14S,  Pvoh, —  To  determine  whether  a  plane  curve  has  rectilinear 
asymptotes. 


Solution.  —First  determine  whether  the  curve  has  infinite  branches.     If  it  has 
not  an  infinite  branch  it  cannot  have  an  asymptote,  since  an  asymptote  is  a  tangent 

Second,  if  there  is  an  infinite  branch, 

dx 


at  an  infinite  distance  from  the  origin 

determine  the  values  of  the  intercepts  of  the  axes  by  a  tangent,  X 


7/-r-,  and 


104 


PROPERTIES   or   PLANE   LOCI. 


Y  =y  —  x^,  for  x  ox  y=.  cc.     It  wiU.  be  necessary  to  observe  wlietlier  both  of 

the  variables,  or  only  one  of  them,  vary  continuously  to  infinity,  and  get  the  value 
or  the  intercepts  in  terms  of  that  one  which  does  vary  continuously.  If  now  one 
or  both  of  the  intercepts  thus  evaluated  is  finite,  the  branch  has  an  asymptote. 
If  both  intercepts  are  infinite,  the  curve  has  no  asymptote,  since  the  tangent  at  co 
does  not  pass  within  a  finite  distance  of  the  origin. 

Having  ascertained  that  the  branch  which  is  being  examined  has  an  asymptote, 
it  remains  to  determine  its  position.  If  the  intercepts  are  both  finite  and  not  0, 
their  values  fix  the  position  of  the  asymptote.  If  one  intercept  is  finite  and  the 
other  infinite,  the  asymptote  is  parallel  to  that  axis  on  which  the  intercept  is  infi- 
nite.    Finally,  if  the  intercepts  are  0,  i.  e.,  if  the  asymptote  passes  through  the 

dy 
origin,  its  direction  is  determined  by  evaluating  —  for  a  tangent  at  infinity. 

ax 

Ex.  1.  Examine  y^  =  6x^  +  ^^  ^or  asymptotes. 

Solution. — Since  as  x  increases  from  0  positively,  y  increases  continuously  and 
■without  hmit,  is  positive  and  has  but  one  real  value,  there  is  an  infinite  branch 
extending  in  the  first  angle.  Now  when  x  is  —,  we  have  y^  =  6x-  —  x"^,  which 
gives  positive  values  to  y  till  x"^  =  &x'^.  After  x^  ^  6x'\  that  is  after  ic  ^  6  and 
negative,  y  becomes  negative  and  a  branch  is  found  extending  in  the  3rd  angle  to 
infinity.  Either  of  these  branches  may  have  an  asymptote,  they  may  both  have 
the  same  line  for  an  asymptote,  or  they  may  have  different  asymptotes.  To  deter- 
mine what  the  facts  are  we  find  the  intercepts  made  by  a  tangent. 
2/2  4a;-^  _|_  a;3  —  2/3        4x'^ -^  x^  —  Qx^  —  x^         —2x2 


X  =  x 


y-. 


4x  -\-x^ 
=  +   CO  =  —  2  ; 
2.t2 


4:X  -\-  x^ 
and   Y  =  y  —  x 


4dX-\-x^ 

A.x-\-x^ 


4.T  -f-  X- 


They  are, 
which  for 


- ,  which  for  x  =  -\-ccz=2. 


y2 

The  branch 


in  the  first  angle  has  an  asymptote  which  cuts  the 
axis  of  2/  at  2  above  the  origin,  and  the  axis  of  x  at 
2  on  the  left  of  the  origin.  The  equation  of  this 
line  is  ?/  r=r  ic  -f-  2.  Finally,  as  the  intercepts  have 
the  same  values  for  ic  =  —  oc  as  for  x  =^  -\-  cc,  this 
line  is  an  asymptote  to  both  branches  of  this  locur,. 
The  curve  is  sketched  in  Fig.  106,  in  which  M  N 
is  the  asymptote. 


Fig.  106. 


Ex.  2.  Show  that  y^  =^  a^  —  x^  has  an  asymptote  which  is  common 
to  its  two  infinite  branches,  passes  through  the  origin,  and  makes  an 
angle  with  the  axis  of  x  of  135°. 

Ex.  3.  Examine  t/2  =  2^  +  3^72  for  asymptotes. 

\  Ex.  4.  Why  has  y^  ■=  x'^  —  x^  no  asymptote  ? 

Ex.  5.  Examine  y^  =  ax'^  for  asymptotes. 

Ex.  6.  Examine  the  conic  sections  for  asymptotes. 


ASYMPTOTES — BY   KECTILINEAR   CO-OEDINATES. 


105 


SoiiUTioN.  — An  ellipse  or  a  circle  can  not  have  an  asymptote  as  neither  has  an 
infinite  branch.     It  remains,  therefore,  to  examine  the  parabola  and  hyperbola. 

From  7/2  =  2px,  we  have  -^  z=  -  ;   hence  X  =  ic  —  ?/—  =  — ^  =  —  ^   which 

is  —  00  for  2/  =  00.     Again  Y  =i  y  —  x-~-  =  y  —  • 


",  which  =  00  for  y  =  oo. 


. ' .  The  parabola  has  no  asymptote.     To  examine  the  hyperbola,  we  have  from 


A-y^-  —  B'^x^' 

=  0  for  X  = 


dy       B^x 


dx 


A^y'-      A"- 


^'^''  ^r,  =  -Ai;, '  ^^^^^  ^  =  ^  -  2/w^.  =  ^  -  -«l7  =  :;' '  ^^^i«^ 


dx      A^y 


±1  GO.     Also  T 


dy 


dy 
B^x'^ 

y-^^y 


B'-x 
B^ 


y 


which  =  0  for 


2/  =  rb  cc,  (In  this  curve  both  x  and  y  vary  continuously  to  infinity,  hence  the 
intercepts  may  be  evaluated  in  terms  of  either.)  .-.  The  hyperbola  has  two 
asymptotes,  and  they  both  pass  through  the  centre.     To  determine  the  direction 


of  the  asymptotes  we  evaluate  -^ 


B'^x 
A^y 


Bx 


fl;  =  4-  CO, 


Bx 


B     ^ 
--=.-.     For  X 

A 


AVx^ 
Bx 


for  a;  =  -+- 


A^ 


For 


B 


=r  = -..      Whence   we 


Fig.  107. 


AV'x-^  —  A^      ^  As/x'^ 

learn  that  the  asymptotes  are  the  produced  diagonals  of  the  rectangle  described 
upon  the  axes,  as  has  been  stated  before. 

IdO,  ScH.  1. — If,  at  successive  points 
along  an  infinite  branch  of  a  curve,  we 
draw  tangents,  these  tangents  wdll  either 
approach  some  limiting  position,  or  they 
will  not.  In  the  hyperbola.  Fig,  107,  it 
is  evident  that  the  successive  tangents 
PX,  P'X',  FT'  are  approaching  the 
limiting  position  SA.  But  in  the  par- 
abola, Fig.  108,  it  is  equally  evident, 
from  the  way  in  which  the  tangents  are  drawn,  that 
there  is  no  limiting  position  beyond  which  a  tangent 
may  not  pass.  Since  AT  =  AD,  AT'  =  AD', 
AT'  r=:  AD",  the  point  of  intersection  wdth  the 
axis  recedes  indefinitely  as  the  point  of  tangency 
passes  to  the  right.  In  a  similar  manner  observing 
the  method  of  drawing  a  tangent  to  an  hyperbola, 
Fig.  102,  it  will  appear  that  as  P  recedes,  the  inter- 
section L  constantly  (but  more  and  more  slowly) 
api^roaehes  E  but  can  never  pass  it ;  and,  consequently,  that  T  can  never 
pass  A,  however  far  P  may  recede.  From  these  considerations,  an  asymp- 
tote is  seen  to  be  tJie  limiting  position  toward  lohich  a  tangent  approaches 
as  the  point  of  tangency  recedes  to  infinity. 

14:7 •  ScH.  2. — Having  found  the  intercept  on  the  axis  of  ordinates  and 
the  tangent  of  the  angle  which  the  asymptote  makes  with  the  axis  of 
abscissas,  we  can  readily  write  the  equation  of  the  asymptote  by  substitu- 


Fig.  108. 


106 


PROPERTIES   OF  PLANE   LOCI. 


ting  my  =  ax  +  b.     Thus  the  equations  of  the  asymptotes  to  the  hyperbola 

B  B 

are  y  —  —x,  and  y  = -x;  or  Ay  =  Bx,  and  Ay  =  —  Bx. 


A 


a 


Ex.  7.  Show  that  y  =  —  x-\--  is  the  equation  of  the  asymptote 

o 

to  ?/3  =  ax^  —  x\ 

Ex.  8.  Show  that  x  =  2a,  and  y  ==  ^  (^  +  a)  are  asymptotes  to 
y^-{x  —  2a)  =  x^  —  a\ 

Ex.  9.  Show  that  the  axis  of  abscissas  {y  =  Ox)  is  an  asymptote  to 
y(a'>  +  ^2)  =  a^[a  —  x). 

Ex.  10.  Show  that  ^  =  2a  is  the  asymptote  to  the  cissoid  of  Diodes. 

Ex.  11.  Examine  x  =  log  y  for  asymptotes. 

Ex.  12.  Examine  y  =  tan  x  for  asymptotes.   -  y-  "^  _'vv 

Sug's. — As  this  curve  is  not  continuous  in  the  direction  x  =  oo,  vre  evaluate  the 
intercepts  for  ?/=  oo,  for  which  x  =  iTt,  f^r,  j7t,  etc.,  and — ^Tt,  — ^Tt,  — |;f,  etc., 

-^  =  sec2£C  =  1  -f-  2/2.     .'.  X  =  x  —  -— y — -.  which  for  y=  cc,=x  =  Itt,  f  ;r,  f;r, 
dx  1  +  y- 

etc,  and  —Iti:,  — Itt,  —^7t,  etc.      Y  =  y  —  x{l  +  y%  which  for  y  =  oo,  =  oo. 

Hence  there  are  an  infinite  number  of  asymptotes  parallel  to  the  axis  of  y.     (See 

23,  M.  27,  Sen.,  Fig.  IS.) 


Ex.  13.  Examine  y==-. 


a3 


-  -\-  c  for  asymptofe?',. 


{x  —  hy 

Sug's. — In  examining  this  locus  it  is 
necessary  to  evaluate  the  intercepts  both 
for  ic  =  00,  {1  nd  i/  =  oo,  as  there  are  infinite 
branches  running  in  both  directions.  The 
general  form  of  the  locus  is  given  in  the 
figure.  The  equations  of  the  asymptotes 
are  a-  =  h  (the  hne  M  N),  and  y  =  c  (the 
line  M'N'). 


14:8,  ScH.  3. — In  very  many  cases 
there  are  more  expeditious  methods  than 
the  above  for  finding  asymptotes.  It  is 
frequently  the  case  that  a  simi)le  inspec- 
tion of  the  equation  of  the  curve  will 
determine  the  fact.  Thus,  in  the  last 
example,  ifc  is  evident  that  as  x  increases 

from  0  to  &,  y  increases,  and  when  x  =  h,  y  becomes  oo.  .*.  This  branch 
of  the  curve  approaches  the  line  MN,  parallel  to  the  axis  of  y,  and  at  a 
distance  h  from  it,  under  the  law  required  for  an  asymptote.  So  again 
when  X  passes  .-r  =  ft,  it  is  evident  that  y  grows  less,  and  the  curve  approaches 
the  axis  of  x.     But,  as,  when  .r=  ck,  y  =  c,  this  branch  extending  to  the 


\ 


TANGENTS  TO   I'OLAR  OUEVES.  107 

riglit  can  never  come  nearer  the  axis  of  x  than  y  =■  c.  In  like  manner  when 
x=^  —  00,  we  see  that  y  =  c     .  • .   M  N '  is  an  asymptote. 

Ex.  14.  Determine  by  inspection  the  asymptotes  to  xy  =  m. 

Ex.  15.  Determine  by  inspection  that  x=^a,  and  y==b,  are  asymp- 
totes to  xy  —  ay  —  hx  =^  0. 

bx  .  ay 

Stjg, — Observe  that  w  ==  ,  and  x  = t. 

•^       X  —  a  y  —  b 

14:9,  ScH.  4. — An  elegant  method  of  examining  for  asymptotes  consists 
in  expanding  y  =/{x),  or  x  =/{y),  into  a  series,  by  the  binomial  theorem, 
Maclaurin's  formula,  or  some  other  method,  when  such  development  is 
practicable.     This  will  be  best  illustrated  by  an  example  or  two. 

Ex.  16.  Determine  the  asymptotes  of  the  locus  x^  —  xy^  +  ay^  =  0, 
by  developing  y  =f(x). 

a 

Now,  if  we  take  the  first  two  terms  of  this  development  we  have  y  =  dz  x  zh  -x, 

the  equations  of  two  straight  lines.  Comparing  this  value  of  ywith  the  entire 
value,  which  is  the  ordinate  of  the  curve,  we  see  that  as  x  increases  the  terms  fol- 
lowing —  grow  less  and  less,  and  consequently  that  the  ordinate  of  the  right  line 
and  the  corresponding  ordinate  of  the  curve  become  more  and  more  nearly  equal ; 
that  is,  the  curve  is  constantly  approaching  the  lines  ?/  =  db  ."^  ±  jr.     Now  when 

a;  =  00  this  difference  vanishes,  as  all  the  terms  following  — -,  become  0.     .  • .  The 

lines  y  =  dzXzh  ha  are  such  that  the  given  curve  approaches  them  constantly  but 
reaches  them  only  at  an  infinite  distance,  and  are  therefore  asymptotes.  There  is 
also  an  asymptote  at  cc  =  a,  which  can  be  discovered  by  inspection,  as  under  the 
last  scholium. 

Ex.  17.  Show  by  developing  y  =^f{x),  that  y  =  =b  ^  are  asymp- 

totes  to  ?/-  =  X- — . 

Ex.  18.  Sbow  by  developing  y  =  /{x),  that  y  =  dz  (x  -{-  a)  are 

X'^  _L  cix" 

asymptotes  to  y"  = . 

Sua. — The  value  of  y  developed  becomes  y=±  x{l  -j \ \-,  etc.). 


(5  )  TANGENTS  TO  POLAE  CURVES. 

150.  It  is  found  most  convenient  to  determine  tangents  to  polar 
curves  by  means  of  the  subtangents. 

151.  Def. — Tlie  Suhtangent  to  a  ^olar  Curve  is  the 


108 


PROPERTIES   OF   PLAKE  LOCI. 


distance  from  the  pole  to  the  ,  tangent,  measured  on  a 
perpendicular  to  the  radius  vector  to  the  point  of 
tangency.  Thus  in  the  figure  let  M  N  be  a  curve 
referred  to  P  as  its  pole,  S  any  point  in  the  curve, 
and  RT  tangent  at  S-  Then  PT  drawn  through 
the  pole,  perpendicular  to  PS  and  hmited  by  the  tan- 
gent, is  the  subtangent. 


152.  J^rop, — The  general  value  of  the  subtangent  to 
a  polar  curve  is 


Subt.  = 


r'^dd  ^ 
Ir   ' 


m 


which  r  is  the  radius  vector  and  0  the  variable  angle. 


Fig.  110. 


Pem. In  tlie  last  figure  let  R  be  a  point  on  the  curve  consecutive  with  S  (infi- 
nitely near  it),  so  that  the  tangent  RT  is  to  be  considered  as  coinciding  with  the 
curve  between  R  and  S.  Draw  PR,  and  with  radius  vector  PS  as  a  radius 
draw  arc  SQ,  and  also  with  radius  Pb  =  1,  draw  ah  an  arc  of  the  measuring 
circle.  Then  RQ  =dr,  since  RQ  is  an  infinitesimal  increment  of  the  radius 
vector,  contemporaneous  with  RS.  So  also  RPS,  or  ab  =  dO.  As  SQ  is 
infinitesimal  it  may  be  considered  a  right  line,  and  it  is  perpendicular  to  PR. 
Again,  as  R  approaches  S,  the  triangle  RQS  approaches  similarity  with  SPT  ; 
and  as  it  is  the  relation  at  the  limit  that  we  seek,  we  are  to  treat  RQS  as  similar 
to  SPT.  Hence  we  have  PT  :  PS  :  :  QS  :  QR,  or  subt.  :  r  : :  QS  :  dr. 
But  from  the  similar  sectors  QPS  and  aPb.  we  have  QS  =  rdQ,  and  substitu- 

r~dQ 
ting,  subt.  :r::rdB  :dr.     . • .  subt  =  -^.     Q-  e.  d. 

Ex.  1.  Find  the  value  of  the  subtangent  to  the  spiral  of  Archim- 
edes. 


Solution. — The     equation     is     r  =  ^7-  ; 


whence    —  =  27r. 
dr 

27r  =  7-7  X  27r  =  — . 


Subt  =  ^^- =r'  X 
dr 


iLii. — The  annexed  figure  furnishes  an  il- 
lustration. PT  is  subtangent  for  point  S 
and  =  the  square  of  the  numerical  value  of 
6  divided  by  the  circumference  of  the  circle 
whose  radius  is  PB.  The  numerical  value 
of  0  in  this  instance  2|7r,  since  in  passing 
from  9  =  0  to  S  the  radius  vector  makes  1§ 
revolutions.       But    one    revolution   =  2;r. 


Fig.  111. 


ASYMPTOTES  TO  POLAR  CURVES. 


109 


Hence  PT  = 


27t,  or  somewhat  more  than  I4  times  the  circumference 


(247r)?_81 

of  the  measuring  circle.  If  B  is  the  point  of  tangency,  0  =t  27t,  and  subt  PT'  = 
27t,  or  the  circumference  of  the  measuring  circle.  In  this  spiral  the  subtangent 
varies  as  the  square  of  the  measuring  arc. 

Ex.  2.  Prove  that  in  the  hyperbohc,  or  reciprocal  spiral  the  sub- 
tangent  is  constant.  "What  is  itsValue,  and  what  the  significance  of 
its  sign  ?  Construct  the  curve  and  tangents  at  a  few  points,  and 
observe  the  subtangents. 

Ex.  3.  Find  the  subtangent  to  the  logarithmic  spiral  (r=  a  ),  and 
show  that  the  angle  under  which  the  curve  meets  the  radius  vector 
is  constant. 

Sttg. — The  tangent  of  the  angle  made  by  a  tangent  to  the  curve  and  the  radius 
vector  is  equal  to  the  subtangent  divided  by  the  radius  vector.  In  this  locus  the 
tangent  of  the  required  angle  is  the  modulus  of  the  system  of  logarithms  used  in 
constructing  the  spiral. 


Ex.  4.   Prove  that  -- 


-r3 


a-*  sin  20 
lemniscate  of  Bernouilli. 


is  the  value  of  the  subtangent  to  the 


153,  I^Tob, —  To  test  polar  curves  for  rectilinear  asymptotes. 

Solution. — Any  curve  which  continually  revolves  around  the  pole  can  have  no 
rectiUnear  asymptote  ;  for  with  respect  to  any  fixed  right  line,  such  a  curve  will 
alternately  approach  and  recede.  But  if  for  some  finite  value  of  0,  r  becomes 
infinite,  the  curve  ceases  to  revolve  around  the  pole,  and  will  have  an  asymptote 
if  the  tangent  at  r  =  co  passes  within  a  finite  distance  of  the  pole  ;  i.  e.,  if  the 
subtangent  is  finite,     q.  e.  d. 

To  construct  the  asymptote,  we  observe  that  the  asymptote  and  radius  vector  are 
drawn  from  a  point  infinitely  distant,  to  the  extremities  of  a  finite  subtangent, 
and  hence  are  to  be  considered  parallel.  We  therefore  determine  the  value  of  the 
subtangent  for  r  =  00,  and  drawing  the  radius  vector  for  that  value  of  6  which 
renders  r  =  co,  erect  the  subtangent,  and  through  the  extremity  remote  from  the 
pole  draw  a  line  parallel  to  this  radius  vector.     This  line  wiU  be  the  asymptote. 

Ex.  1.  Test  the  hyperbola  for  asymptotes  by  the  polar  method. 

Solution. — The  polar  equation,  when  the 

^(1  —  e2) 


pole  is  at    F,  is  r  = 


e  cos  Q  —  1' 


Now  for 


cos  Q=-,r=  00  ;  hence  if  there  be  an  asymp- 
FR    so   drawn    that 
From  the  equa- 


tote,  it  is  parallel  to 

cos  R  FX  =  -  =  — . 


d9 
tion  we  have  -7-  = 

C4/- 


(e  cos  0  — 


Ae{l  —  e^)  sm  0 


—  ;   whence 


Fig.  112. 


110  PEOPERTEES  OF  PLANE  LOCI. 

dQ  ^2(1— e2)2  (e  cos  0  —  1)2  Ail  —  e^)      ,  .  ,       .         - 

suht.  =  r-—  =  ^-7 X  -rr—. ^— — t:  =  ^— 7r->  which,  since  for 

dr        (e  cos  0  —  1)2  ^  Ae{l  —  e^)  sm  0  esm  0    ' 

cos  0  =  -,  sin  0  =  -  s/e^  —  1,  =  —  A\/e-  —  1  =  —  B.     There  is  therefore  an  asymp- 
e  e 

tote.     To  construct  it  draw  FR  making  RFX  =  cos-M-V  FD  perpendicular 

to  R  F  and  =  B,  and  through  D  draw  SX  parallel  to  R  F.  ST  is  the  asymp- 
tote. Moreover,  since  cos(— 0)  =  cos0.  there  is  another  asymptote  similarly 
situated  below  the  polar  axis  FX.     Finally,  as  the  angle  which  a  diagonal  upon 

the  axes  of  an  hyperbola  makes  with  the  axis  of  x  is  cos— * —  .  the  asymp- 

VA^  -\-  J?2 

totes  are  parallel  to  these  diagonals  ;  and  since  F  D  =  ^,  A  F  =  \/A-  -j-  B^,  and 
the  asymptotes  coincide  with  the  diagonals. 

Ex.  2.  Show  by  the  polar  method  that  the  parabola  has  no  asymp- 
tote. 

SuG.  — In  this  case  for  0  =  0,  r  =  oo  ;  but  for  this  value  of  9  suht  =  oo.     Hence 
there  is  no  asymptote. 

Ex.  3.   Show  that  the  hyperbolic  spiral  has  an  asymptote  parallel 
to  the  polar  axis  and  at  a  distance  27t  from  it. 

Ex.  4.  Show  that  the  polar  axis  is  an  asymptote  to  the  lituus. 


-4-»">- 


SECTION  11. 
Normals  to  Plane  Loci. 

(a)    BY  KECTANGULAE  CO-OKDINATES. 

'154z»  Def. — A  Novinal  to  a  plane  curve  is  a  perpendicular  to  a 

tangent  at  the  point  of  tangency. 

1^5.  I^TOp, — The  general  equation  of  a  normal  to  a  plane  curve  is 

dx' 

y  —  y=  —  j-,{^  —  ^')> 

in  which  (x',  y')  is  the  point  in  the  curve  to  which  the  normal  is  drawn, 
and  X  and  y  are  the  general  co-ordinates  of  the  normal. 

Dem. — Letting  [x  ,  y')  represent  the  point  in  the  curve  to  which  the  normal  is 

dy'  , 

to  be  drawn,  the  equation  of  a  tangent  through  this  point  is  y  —  y'  =  -f-:i{^  —  ^  )• 

Again  the  equation  of  any  line  passing   through  (a;',  y')  is  y  —  t/'  =  a(x  —  x). 
Now,  in  order  that  the  equation  for  this  general  line  should  become  the  equation 


NORMALS — BY   RECTANGULAK   CO-ORDINATES. 


Ill 


of  a  perpendicular  to  the  tangent,  a  must  = 
is  the  equation  of  a  normal,     q.  e.  d. 


dx,' 
dy' 


y  —  y 


dx 
dy 


{X—  X) 


156.  Cor. — The  general  expression  for  the  tangent  of  an  angle  vMch 

dx 
a  normal  makes  with  the  axis  of  abscissas  is  —  —- ,  (x,  y)  being  the  point 

.f  ^y 

~  in  the  curve  to  which  the  normal  is  drawn. 

Ex.  1.  Produce  the  equations  of  the  normals  to  the  conic  sections. 


Results,  Ellipse,  y  —  y' 


A^y' 


y 


^l^(^  —  ^0 ;  Circle,  2/  =  ^,^ ; 


A^y'  y' 

Hyperbola,  y~y'=—^/-,{x—x');'Parahola,y—y'=—^-(x—x'). 

^  X  p 

ScH. — Observe  that  these  equations  do  not  reduce  to  as  simple  and  sym- 
metrical forms  as  do  those  of  the  tangents  to  the  conic  sections.  The  form 
of  the  equation  of  the  normal  to  the  circle  shows  that  the  normal  of  this 
locus  always  passes  through  the  centre.     It  is,  of  course,  the  radius. 

Ex.  2.  What  is  the   equation  of  the  normal  to  y^  =  2x^ x^  at 

x  =  l? 

Answer  :  At  x=l,  y=dr.l ;  hence  there  are  two  points  indicated. 

The  equation  of  the  normal  at  the  former  is  y  =  2x  —  3,  and 

at  the  latter  y  =  —  2a;  -f  3. 

Ex.  3.  What  is  the  equation  of  a  normal  to  t/^  =  6.r  • —  5  at  2/  =  5  ? 
What  angle  does  this  normal  make  with  the  axis  of  ^  ? 

Ex.  4.  At  what  point  in  the  ellipse  whose  axes  are  12  and  8  must  a 
normal  be  drawn  to  make  an  angle  of  45°  with  the  axis  of  ^? 

Ex.  5.  At  what  point  in  the  witch  of  Agnesi  must  a  normal  be 
drawn  to  be  perpendicular  to  the  axis  of  a;  ?  To  be  parallel  ?  To 
make  an  angle  of  135°  ? 


157.  'Def. — The  Subnormal  is  the.  projection  of  the  normal 
upon  the  axis  of  x ;  or  it  is  the  distance  from  the  foot  of  the  ordinate 
let  fall  from  the  point  in  the  curve  to  which  the  normal  is  drawn,  to 
the  intersection  of  the  normal  with  the  axis  of  x. 

158.  J^Toh. — To   find  the  general 
value  of  the  subnormal. 

Solution. — In  Fig.  113  P  E  is  the  normal  and 

D  E  the  subnormal  for  the  point  P.     Now  in 

the  triangle  P  D  E,  P  D= y,  and  tan  P  E  D  = 

dx 
(numerically)  tan  PEX  = 


by  trigonometry. 


dy 


-  °-=4 


Fio.  113. 


112 


PEOPERTIES   OF  PLANE  LOCI. 


TA  D 


Fig.  114. 


Ex.  1.  Show  that  the  subnormal  to  the  parabola  is  constant  and 
equal  to  half  the  latus  rectum.  How  can  a  tangent  be  drawn  to  the 
parabola,  geometrically,  upon  this  principle  ? 

\    Ex.  2.  Show  that  the  subnormal  to  the  cycloid  is  {2ry  —  y^)  . 

159.  Con.— Since  DC  =  PG 
=-v/CG  X  GS  =  \/y(2r  — y)"  = 
V  2ry  —  y2,  tlie  normal  passes 
through  the  foot  of  the  vertical  diam- 
eter of  the  generating  circle  for  the 
point  to  which  the  normal  is  drawn. 
Moreover^  since  SPG  is  a  right  angle,  the  tangent  passes  through  the 
other  extremity  of  the  vertical  diameter. 

160.  ScH,  1. — This  principle  affords  a  ready  method  of  constructing  a 
tangent  to  the  cycloid  geometrically.  Let  P  be  the  given  point  through 
which  a  tangent  is  to  be  drawn.  Put  the  generating  circle  in  position  for 
this  point  {94),  and  draw  the  vertical  diameter  SC  Through  S  and  P  draw 
a  right  line  and  it  will  be  the  required  tangent.  Also  PC  wiU  be  the 
normal  to  the  curve  at  the  point  P. 

161.  ScH.  2. — To  draw  a  tangent  which  shall  make  any  given  angle  with 
the  axis  of  x,  draw  the  generating  chcle  on  the  axis  HI,  construct  the  angle 
LHI  =  the  complement  of  the  required  angle,  and  through  L,  the  point 
where  this  line  intersects  the  circumference  of  the  central  generating  circle, 
draw  a  parallel  to  the  base  of  the  cycloid.  "Where  this  parallel  cuts  the 
curve  P  is  the  required  point  of  tangency.  Tlu'ough  this  point  draw 
SPX  parallel  to  HL,  and  it  is  the  tangent  required. 


162,  J^roh. — To  find  the  length  of  the  normal,  i.  e.,  the  portion 
intercepted  between  the  curve  and  the  axis  of  x. 

Solution. — In  Fuj.   113,  from  the  right  angled  triangle,  PDE  we  have  PE  = 


V 


*/• 


PD-4-DE-  =  j2/>  +  2,=-=2,      1+-.     Q.E.D. 


4 


dif~ 


Ex.  1.  Find  the  length  of  the  normal  in  each  of  the  conic  sections. 
What  is  it  in  the  circle  ? 

K    Ex.  2.  In  the  cycloid  the  radius  of  whose  generatrix  is  2,  what  is 
the  length  of  the  normal  at  ?y  =  1  ?  Ans.,  2. 

TGS,  Cor. — The  normal  is  hut  a  particular  case  of  a  perpendicular 
to  a  tangent. 


NORMALS   TO   POLAR   CURVES. 


113 


Dem. — ^As  y  —  y'  =  -r^{^  —  ^')  is  the  equation  of  a  tangent  at  (x',  y'),  a  perpen- 


dx 


dx' , 


dicular  to  this  through  the  point  {x",  y")  is  y  —  y"  = j-,ix  —  x").     Making 

cly 

the  point  through  which  this  perpendicular  to  the  tangent  is  to  pass  the  point  of 

tangency,  the   perpendicular  becomes  a  normal,   and  its  equation   is  2/  —  y'  == 


dx 
dy 


-{X  —  x'),  since  in  this  case  x"=  x  ,  and  y"  =  y  .     For  a  perpendicular  from 

dx' 


the  origin  on  the  tangent  we  have  2/"  =  0,  a;"  =  0,  and  y  = —x. 

Ex.  1.  Show  that  the  equation  of  a  perpendicular  from  the  focus 


V 


of  the  common  parabola  upon  the  tangent  is  ?/  = {^x  —  ^jy). 

P 
Ex.  2.  Show  that  the  perpendicular  distance  from  the  focus  of  an     / 
hyperbola  to  the  asymptote  is  B. 

104:,  CoR. — The  perpendicular  from  the  focus  of  a  parabola  upon 
a  tangent  meets  the  tangent  in  a  tangent  to  the  curve  at  the  vertex  (the 
axis  oi  y). 

Dem. — The  equation  of  a  tangent  is  yy'  =  p{x  -\-  x),  and  of  the  perpendicular 

v' 

from  the  focus  upon  this  tangent  2/  =  —  —{x  —  Ip).  "We  have  now  but  to  find  the 
intersection  of  these  hues.     Equating  the  values  of  y,  we  have  —  —  (x  —  ip)  = 

—{x  -f-  ic'),  or  —  y'^{x  —  ip)  =  p-jc  -j-  p^^\  01^  since  y'^  =  2px',  —  2pxx'  -f-  p^^'  == 

p^o; -f-p2/);' ;  whence  {p-\-2x')x  =  0.  Now  as  x'  can  not  be  — ,p-^2x'  can  not 
become  =0.  Therefore  to  fulfill  the  condition  {p  -^2x')x  =  0,  x  must  =s  0. 
.• .  The  point  of  intersection  is  at  ic  =  0,  or  in  the  axis  of  y.     Q.  e.  d. 


(b)  NOEMALS  TO  POLAR  CURVES. 

16S,  Def. — The  Subnormal  to  a  polar  curve 
is  the  distance  from  the  pole  to  the  normal,  measured 
on  a  line  perpendicular  to  the  radius  vector  to  the 
point  in  the  curve  to  which  the  normal  is  drawn. 
Thus  E  P  is  the  subnormal  of  the  curve  M  N  corres- 
ponding to  R,  the  pole  being  at  P. 

100,  I^TOh, — To  find  the  general  value  of  the  sub- 
normal to  a  polar  curve. 


Solution.     PT  =  r 


But  tanPER-*^= 
PR 


^d6 
dr 


PR 

Tan  PTR  =  -^  =  -—  = 


PT 


ifZ. 


r 
~de 


dr 

rdff 


dr 


tan  PER 


dr_ 


tan  PTR 

Q.  E.  D. 


rdB 
dr' 


=  -r-.     .'.  P E  or  subnormal  = 


Fig.  115. 


114  PKOPERTIES  OF  PLANE  LOCI. 

Ex.  1.  Show  that is  the  value  of  the  polar  subnormal 

r 

of  the  lemniscate  of  Bernouilli. 

T 

Ex.  2.  Show  that  the  subnorraal  to  the  logarithmic  spiral  is  — ,  7n 

being  the  modulus  of  the  system  ;  and,  consequently'',  that  in  the 
Napierian  logarithmic  spiral  the  subnormal  always  equals  the  radius 
vector. 

(dr^         \i 
—  +  r^j  . 

IGS,  CoR.  2. — The  length  of  a  perpendicular  from  the  pole  upon  a 

r2 
tangent  is  p  = 


/dr 
Vd^ 


Dem. — In  FiQ.  115,  let  PS  be  the  perpendicular  from  the  pole  upon  the  tangent, 
and  consequently  parallel  to  the  normal  R  E.     From  the  right  angled  triangle 

PST,   PS  =  PT  X  cosSPT.      But  PT    (the  subtangent)  =  -j^  \    and 

cos  SPX  =  cos  RET  =  ftf^^^  = 1  =  „  .c^^  ,- 

sec  RET  ^     ,    X     .„^_^xa        /^     ,    r^dB^\k 


(l  +  tan-3RET)^        {}  + -^) 


rHQ 


Whence  p  = — — :  =  — -.     q.  e.  d. 


0+'1?)'  C^.+-)* 


■#♦  » 


SUCTION'  IIL 

Direction  of  Curvature. 

{a)  BY  KECTANGULAB    CO-OEDINATES. 

d^Y 
ISO*  JPvop.—At  a  point  where  -^  is  positive,  a  curve  is  concave 

d^Y  .  , 

upward,  and  where  j-^  is  negative  the  curve  is  convex  upward. 

Dem. — 1st.  Let  go  be  the  angle  which  a  tangent  makes  with  the  axis  of  x.  When 
the  curve  is  concave  upward,  as  in  B.g.  116,  it  is  evident  that  as  x  increases  (as  from 
being  the  abscissa  of  P  to  being  that  of  P' ),  oa  increases.  In  other  words,  if  x  takes 
the  infinitesimal  increment  dx,  the  contemporaneous  infinitesimal  change  in  ca  is 


DIRECTION   OF   CURVATURE. 


115 


-f-  dao.  Hence  when  the  curve  is  con- 
cave upward,  dx  and  doo  have  the  same 
sign  (a;  and  go  are  increasing  functions 
of  each  other). 

In  a  similar  manner  it  is  evident  that 
when  the  curve  is  convex  upward,  go 
decreases  as  x  increases  ;  i.  e.,ii  x  takes 
the  increment  dx,  the  cortemporaneous 
change  in  co  is  — dca. 

dy 

dx        d  '  tan  go 


2nd.  P, 

dx^ 

doo 


d 


dx 


dx 


Bec2  GO— :      Now,   as  sec"  go  is  always 

(J/*C 

positive,   -T^  is  positive  when  x  and  go 

are  increasing  functions  of  each  other 
(when  dx  and  dGo  have  like  signs),  and 
negative  when  they  are  decreasing  func- 


tions of  each  other. 


-f  -r-  mdi- 
dx^ 


cates  that  the  curve  is  concave  upward,  and 
upward,     q.  e.  d. 

Another  Demonstration. — Let  DD',  and  D'D", 
Fig.  118,  represent  consecutive  equal  infinitesimal 
increments  of  x,  then  P'E  and  P  E'  represent  con- 
temporaneous infinitesimal  increments  of  y.  Eepre- 
sent  them  respectively  by  dyi  and  J?/2.  The  differ- 
ence between  di/^  and  dy2,  is  by  definition  d^y. 
Bat  when  a  curve  is  concave  upward  it  lies  above 
its  tangent.  Hence  dy2  >>  dy-,  and  dyz  —  %i  = 
-\-  d-y.  On  the  other  hand,  when  the  curve  is  con- 
vex upward,  as  in  Fig.  119,  it  lies  below  its  tangent 
and  dy2  <idi/-[.  "Whence  dy.j.  —  dyi  =  —  d-y.  A 
similar  inspection  can  easily  be  made  in  all  cases, 
both  when  the  curve  lies  above  the  axis  of  x,  and 
when  it  lies  below,  an4.  thus  the  universality  of  the 


Fig.  117. 

-^  indicates  that  it  is 
daifl 


convex 


Fig.  118. 


principle  be  established.     Finally,  the  sign  of 


d^ 
dx^ 


is  the  same  as  that  of  d"y,  since  dx-  being  a  square 
is  always  positive. 

170,  Cor.  1. — By  a  course  of  reasoning  en- 
tirely similar^  taking  y  as  the  independent  vari- 

77         .  7  ,  7  ^"^ 

able,  it  may  be  shown  that  -f 


t/ 

Y 

P 

; 

R 

E 

/* 

[ 

3    [ 

3'  C 

)"              X 

d^x 


dy 


Fig.  119. 
indicates  that  a  curve  is  concave 


to  the 


riaht,  and r-  that  it  is  convex  to  the  right. 

^    '  dy^ 


116  PROPEKTIES  OF  PLANE  LOCI. 

17 !•  Cor.  2. — A  curve  is  convex  towards  the  axis  of  abscissas  when 
y  Y^  is  positive,  and  concave  when  it  is  negative. 

Dem.— For  points  above  the  axis  of  x,  y  is  -f ,  and  if  tlie  curve  is  convex  towards 
the  axis  (downward)  -rf  is  also  -|-  ;  hence  y~^  is  -}-.     For  points  below  the  axis 

y  is  — ,  and  if  the  curve  is  convex  towards  the  axis  (upward)  —  is  — ;  hence 

dfiv  d'tj 

II— ^  is  4-.     Therefore  ?/— ^  is  always  4-  when  the  curve  is  convex  towards  the  axis 

dHi 
of  X.     In  a  similar  manner  it  may  be  seen  that  y-r^  is  —  when  the  curve  is  con- 
cave towards  the  axis  of  x. 

Ex.  1.    To  discover  whether  x^  -\-  y^  =  r-  is   convex  or  concave 
towards  the  axis  of  x. 

d^v  T^ 

SoiiUTioN. — From  x^  -{-  y^  =^  r%  we  have  ^-^  = .     This  locus  is,  therefore, 

dx^  y"^ 

convex  upward  when  y'\&-\-,  and  concave  when  y  is  — .     Hence  it  is  always  con- 
cave to  the  axis  of  x. 

Ex.  2.   Test  the  following  for  direction  oL  curvature  :    ?/  =  6  + 
c{^x  +  a)2  ;  and  y  =  a:W x  —  a. 

Results,   The  first  is  concave  upward  ;    and  the   second  concave 
towards  the  axis  of  x. 

Ex.  3.  Test  the  direction  of  curvature  y  =  6+  (^  —  o)\ 
Results,   From  ^  ^  a  to  ^  =  oo  convex   towards   the   axis  of  x. 

1  1 

From  x<^aio  x=  a  —  6^,  concave.     From  x=  a  —  h^  io  x  = 

—  00,  convex. 
Ex.  4  Examine  2/  =  sin  a; ;  x  =  log  y  ;  y  =  tan  x. 


(h)  BY  POLAE  CO-OEDINATES. 

172,  Bef. — ^A  Polar  curve  is  said  to  be  concave  or  convex  towards 
its  pole  at  any  point,  according  as  the  curve  at  that  point  does,  or  does 
not,  lie  on  the  same  side  of  its  tangent  as  the  pole. 

173*  JProp. — A   polar  curve  is  concave  tovmrd  the   pole  when 

■r—  is  positive,  and  convex  ichen  -—  is  negative  ;   r  being  the  radius  vector 
dp  dp 

and  ]}  the  jx'rjoendicidarfroin  the  pole  upon  the  tangent. 


SINGULAR   rOINTS. 


117 


Dem. — By  a  simple  inspection  of 
(a)  Fig.  120,  it  will  be  seen  that  r 
and  p  are  increasing  functions  of 
each  other  when  the  curve  and  pole 
lie  on  the  same  side  of  the  tangent ; 


In  like  manner 


(If 

hence    -r-   is   -{-. 

dp 

from  (&)  it  is  seen  that  r  and  p  are 

decreasing  functions  of  each  other 

when  the  pole  and  curve  lie  on  dif- 

dv 
ferent  sides  of  the  tangent ;  hence  --  is 
®  dp 


Fig.  120. 


174:,  ScH. — In  applying  this  polar  test  for  direction  of  curvature,  it  is 
necessary  that  the  equation  be  in  terms  of  p  and  r.     If  given  in  r  and  0, 

6  can  be  eliminated  between  the  equation  of  the  curve,  and  p  = 


{168). 


Ex.  1.  Examine  the  lituus  (  r  =  —  )  vrith  reference  to  direction  of 
curvature. 


Sug's.  —From  r  =  -  ,  37-  =  ia^Q-^  =  ■—.     This  substituted  in  »  = 
A  dQ^  ^'^^  i^ 


4:a'i' 


a+^-y 


gives  p 


la^r 


dr         (4a4  _|_  ^4) 
Whence  --  = 


dp       2a\4:a*  —  r-")' 


.  This  spiral  is  concave 


(r4  _|_  4a'*) 
towards  the  pole  for  values  of  r  less  than  a\/2,  and  convex  for  r  >>  «\/2. 

Ex.  2.  Sliow  r=  a   is  always  concave  towards  the  pole. 


SUCTION  IV, 
Singular  Points, 

17 S.  "Def.— Singular  Points  of  curves  are  points  which 
possess  some  property  not  common  to  others.  Of  such  points  we 
shall  notice  :  1st,  Points  of  maxima  and  minima  ordinates  ;  2nd, 
Points  of  inflexion  ;  3rd,  Multiple  points  ;  4th,  Cusps  ;  5th,  Isolated 
or  Conjugate  points  ;  6th,  Stop  points  ;  7th,  Shooting  points. 


118 


PROPERTIES  OF  PLANE  LOCI. 


MAlTTIffA   AND  MEVIMA  ORDINATES. 

17 6 •  Def. — An  ordinate  is  at  a  maximum  when  it  is  greater  than 
the  immediately  preceding  and  the  immediately  succeeding  values  ; 
and  at  a  minimum  when  it  is  less  than  the  immediately  preceding  and 
immediately  succeeding  values. 

17  7 •  IPvoht — To  find  the  position  and  values  of  maxima  and  min- 
ima ordinates. 


Solution. — As  y=f{x),  this  problem  is  the  ordinary  one  of  maxima  and  minima  of 
functions  of  a  single  variable,  treated  in  the  Calculus.     Hence  we  find  the  values  of  x 

•which  render  —  =  0  or  00,  as  critical  values,  i.  e.,  values  to  be  examined,  and  at 
dx 

whicb  the  property  exists,  if  it  exist  at  all.     To  distingmsh  between  maxima  and 
minima  values  we  have  the  common  test  ;  namely,  -|-  -y-^  characterizes  a  mini- 

mum,  and ~  characterizes  a  maximum,  subject  to  the  conditions  discussed  m 

dx,' 

the  Calculus.     The  value  or  values  of  y  corresponding  to  the  value  or  values  of  x 

found  as  above,  will  be  the  required  maxima  or  minima  ordinates. 

A  Geometrical  Solution. — If  PD 

is  a  maximum,  it  is  evident  that  at 

the  left  of  P  the  tangent  makes  an 

acute  angle  with  the  axis  of  x,  i.  e. 

dv  dv 

--  is  4- ,  and  at  the  right  --  is  — .     .  • . 

dx,  dx 

dy 

--  =  0  is  the  point  of  change  from 

dx 

-f-  to  — ,  or  the  point  of  maximum 

ordinate.      In  hke  manner  at  the  left  of  a  point  of  minimum  ordinate,  as  P', 


_FU__ 
A  D  D»  >? 


dx 


is  — ,  and  at  the  right  +. 


--  =  0  locates  also  minimum  ordinates.* 
dx 


Finally,  since  at  a  point  of  maximum  ordinate  the  immediately  preceding  and  suc- 

dv 
ceeding  ordinates  are  less,  the  curve  is  concave  downward,  whence  we  have  —  -^ 

characterizing  such  a  point.     But,  at  a  point  of  minimum  ordinate,  the  immedi- 
ately preceding  and  succeeding  values  of  y  being  greater,  the  curve  is  convex 

d^y 
downward  and  we  have  -|-  —  characterizing  this  point. 

178,  ScH. — If  only  the  numerical  values  of  the  ordinates  be  considered, 

d^v     .  .  .   .  d^7/ 

when  P  lies  below  the  axis, will  characterize  a  minimum,  and  +  -; — » 

maximum.       But  a  numerical   maximum,  if  — ,  is   properly   considered  a 
minimum  ;  and  a  negative  numerical  minimum,  is  properly  a  maximum. 


dy 


*  ^or  ^  =  CD ,  606  Calculus,  p.  9t 


Ex.  1.  Examine  y 
ordinates. 


SINGULAK  POINTS. — POINTS   OF   INFLEXION.  113 

x^  —  9^2  _^  24^  +  16  for  maxima  and  minima 


Solution. 


dx 


3a;2  —  18a;  +  24  =  0. 


x  =  4.  and  2,  — ^ 
dx^ 


ex  — 18.     For 


d^y 
X  =  4:,  —  =  6  ;  hence  x  =  4:  corresponds  to  a  minimum,  which  value  is  32.     For 

d^y 
X  =  2,  -—  =  —  6  ;  hence  x  =  2  corresponds  to  a  maximum,  which  value  is  36. 

[Note. — The  student  should  construct  the  locus,  and  notice  the  points.  Also  substitute  values 
for  X  a  little  greater  than  4:  and  a  little  less,  and  the  same  for  the  point  x  =2,  observing  in  the 
results  the  maxima  and  minima  values  of  y.  ] 

Ex.  2.  Find  the  location  and  value  of  maxima  and  minima  ordi- 
nates in  the  following  curves  :   [1),  y  =  x'^  —  5x*  +  5x^  +  1  ;  {2),  y  = 

x^  —  Sx^  —  24.x  +  85  ;  (3),  y  =  5{x  —  x^)  ;  (4),  y  =  {2ax  —  x^)^  ; 
{5),y  =  x^  —  8x^  +  22x^~2^x-{-12',{6),y  =  b-\-{x  —  ay;  (l),y  = 
x'^{a  —  xy  \  (8),  in  the  logarithmic  curve  ;  (9),  in  the  curve  of  tan- 
gents ;  (10),  in  the  cycloid  ;  (11),  in  the  parabola  ;  (12),  in  the  lem- 
niscate  of  Bernouilli. 


POINTS   OF   INFLEXION, 

(a)   BY  KECTANGULAB,  CO-OBDINATES. 

170*  I>EF. — A  JPoint  of  Jftflexiou  is  a  point  where  a  curve 
changes  direction  of  curvature  for  continuously  increasing  values  of 
X  or  y.  Such  a  point  is  also  characterized  by  the  fact  that  the  tan- 
gent at  the  point  cuts  the  curve  in  the  point  of  tangency. 

Iiiii. — In  passing  from  P'  to  P",  the  curve 
M  N  changes  direction  of  curvature,  being 
convex  downward  at  P',  and  upward  at  P". 
The  point  P  at  which  this  change  occurs  is  a 
point  of  inflexion.  The  student  should  not 
confound  a  point  of  inflexion  with  such  a 
point  as  P  in  M '  N '.  It  is  true  that  reckon- 
ing along  the  curve  from  M '  to  N '  the  curve 
changes  direction  of  curvature  with  reference 
to  the  axis  of  x  ;  but  not  so  in  reckoning  along 

AX.     From  D'  to  D  the  curve  is  both  concave  and  convex  towards  the  axis,  and 
does  not  change  at  P,  but  is  limited  there. 

ISO,  I*TOh, — To  determine  points  of  iriflexion. 

Solution. — If  examined  with  respect  to  the  axis  of  x,  since,  when  the  curve  is 

d'^y  d^y 

convex  downward  we  have  A — -^,  and  when  concave  downward -,  at  the 

rZx2  cZxs 


Fig.  122. 


120 


PEOPEETIES  OF  PLANE  LOCI. 


point  of  inflexion  —  must  change  sign,  and  hence  must  =  0,  or  oo,     .-.If  there 

be  a  point  of  inflexion  it  is  where  ^  =  0  or  oo.     Having  determined  this  point, 

either  construct  the  curve  in  the  neighborhood  of  it,  or,  better,  substitute  in  ^  a 
value  of  X  a  httle  greater,  and  one  a  little  less  than  the  critical  value,  and  observe 

'^^®^^^''  ^  '■^''^^^  ^""^^  ""^^"^^^  ^^g^  ^^  ^^e  point  under  consideration. 

The  precaution  in  the  latter  part  of  this  solu- 
tion is  necessary  ;  for,  though  a  varying  quan- 
tity cannot  change  sign  without  passing 
through  0  or  oo,  it  does  not  necessarily  change 
sign  upon  passing  through  these  values.  Thus, 
let  M  N  be  a  curve  whose  equation  is  y  ^=f{x). 
Now,  as  X  passes  from  the  value  A  D  to  that 
of  AD',  ?/ passes  through  0,  hut  does  not  change 
its  sign.  In  like  manner  by  referring  to  Fig.  109, 
it  will  be  seen  that  in  the  curve  there  delineated,  y  passes  through  oo  without 
changing  its  sign. 

Ex.  1.  Examine  y  =  6+  (^  —  ciY  for  points  of  inflexion. 
Solution.     -—  =  Q[x  —  a)  =  0,  gives  x^a,  as  a  critical  point,  i.  e.,  one  which 


d^p 


dy 


may  have  the  property  sought.     Now  for  x'y>  a,  ^  is  + :  and  for  o!  <<  a,  ^ 

dx;^  dx- 

is  — .     Therefore  there  is  a  point  of  inflexion  at  x=^a.     For  x  =  a,  y  ^^h  ;  hence 

the  point  of  inflexion  is  (a,  6). 

Ex.  2.  Examine  the  following  for  points  of  inflexion  :  a'^y=,x'i  —  cjc^  : 
y  =  x-\-SGx^  —  2x^  —  £c-i ;  y  =  since;  y=taiix;  a;  =  logy;  the  witch 
of  Agnesi. 


(b)   BY  POLAE   CO-OEDINATES. 
ISl,  IPvoh, — To  test  polar  curves  for  points  of  infiexion. 
Solution. — The  equation  being  put  into  the  formp  =f{r),  we  have  seen  that 


dp 


dp 


for  —   -[-,  the  curve  is  concave  towards  the  pole,  and  for  ~  — ,  it  is  convex. 


dr 


dp 


dp 


Therefore   —  =  0,  or  oo,  indicates  a  critical  point.     If  upon  examination    -=-  is 


dr 


dr 


found  to  change  sign  at  this  point,  the  point  is  one  of  inflexion,     q.  e.  d 
Ex.  1.  Test  the  lemniscate  of  Bernouilli  for  points  of  infiexion. 


Solution. — The  equation  is  r'^  =  a'^  cos  2d  ;  whence  we  have  p 


±a^ 


and 


SINGULAR   POINTS — MULTIPLE  POINTS. 


121 


±  a 


2  ' 


Putting  =  0,  r  =  0.    If,  therefore,  there  is  a  point  of  inflexion 


dr 

it  is  at  r  =  0,  that  is,  at  the  pole.     Finally,  ^  =  ^-^ 


da'^  cos  26 


=  ±  3  cos  26, 


which  changes  sign  for  consecutive,  real  values  of  r  ;  i.  e.,  when  6  passes  from 
45°  to  135°  for  which  change  r  passes  through  0. 


Ex  2.  Examine  the  lituus  (r  =  ~)  for  points  of  inflection. 

There  is  a  point  of  inflection  at  r  =  aV^>  ^  =  ^8°  38'  +  . 

/7/93 

Ex.  3.  Examine  r--^ r  for  points  of  inflexion. 

(7^  i 


Solution.     -^7-  =: 


P 


-,  and  f- 
k  dr 


(4^4  _  I2ar3  -\-  13aV2  —  4.a^r) ' 
(6r^_13ar  +  6a^)(-Q^V)^_^      Whence  r  =  0,  fa,  and  |a.     If,  therefore, 
(4,^4  _  I2ar3  j^  ISaY^  —  4:a^rV^ 
there  is  a  point  of  inflexion,  it  must  be  where  r  passes  through  0,  |a,  or  |a.     But  / 

^  changes  sign  only  with  the  factor  6r2  —  13ar  -f-  Ga^ ;  and  this  factor  does  not 

dr 

change  sign  when  r  passes  through  0,  but  does  at  r  =  |a  and  fa.     (To  determine 

these  facts,  substitute  r  =  Q-^h,  and  r  =  0  —  7i ;  also  r  =  fa  +  ft,,  and  r  =  fa  —  ft, 

etc.,  /t  being  treated  as  infinitesimal.)    .  * .  There  is  a  point  of  inflection  at  r  =  fa. 

Where  r  =  |«,  0  =  V3,  or  about  99°.26.    Where  r  =  |a,  0  is  imaginary. 


MULTIPLE   POINTS. 

182,  Def. — There  are  two  species  of 
Multiple  I^oints^  viz.,  1st,  A  point  where 
two  or  more  branches  of  a  curve  intersect ; 
2nd,  A  point  where  two  or  more  branches  are 
tangent  to  each  other.  The  latter  are  some- 
times called  Points  of  Osculation.  The  an- 
nexed figures  illustrate  both  species.  The 
first  curve  has  a  triple  point  of  the  first  species 
at  P,  and  the  second  a  double  point  of  the 
second  species  at  P. 


Fig.  124. 


183.  JProb. 

muUiple  points. 


■To  examine  a  curve  for 


Fig.  125. 


Solution.  — Since  two  or  more  branches  pass  through  a  multiple  point,  for  x  = 
the  abscissa  of  such  a  point,  y  has  but  one  value,  while  at  other  points  near  it,  y 
has  two  or  more  values  for  each  value  of  x.     In  explicit  functions,  or  in  functions 


122  PEOPEKTIES  OF  PLANE  LOCI. 

of  a  comparatively  simple  form,   such  a  point  can  generally  be  determined  by 
inspection.     Having  found  a  value  of  x  for  wliich  y  lias  but  one  value,  and  on  both 

sides  of  which  it  has  two  or  more,  form  -^,  and  observe  whether  it  has  equal  or 

dx 

dv 
unequal  values  at  this  point.     If  -^  has  unequal  values  the  branches  of  the  curve 

intersect  at  the  point,  since  their  tangents  do,  and  the  point  is  of  the  first  species. 

If  —  has  but  one  value  for  these  values  of  a;  and  y,  the  tangents  to  the  branches  at 
dx 

the  point  coincide  and  the  point  is  of  the  second  species. 

When  the  critical  points  are  not  readily  determined  by  inspection,  put  the  equa- 
tion in  the  form  of  an  imphcit  function  without  radicals.     Let  it  be  u  =f{x,  y)  =  0. 
du 

Form  —  =  —  — .     Now,  as  the  equation  of  the  locus  did  not  contain  radicals, 
dx  du 

dy 

dv 
and  as  differentiation  does  not  introduce  them,  the  only  way  in  which  --  can  have 

du 

0      __  .         dy  dx      0        du 

several  values  is  by  taking  the  form  -.     Hence  we  have  -5-  =  —  7"  =^  n'  ^^  dT^^ 

dy 

and  —  =  0,  from  which  to  determine  critical  values  of  x  and  y,  (that  is,  those 
dy 

du 
values  which  may  correspond  to  multiple  points).     Solving  the  equations  —  =  0, 

and  —  =  0,  for  x  and  y,  see  which  of  the  values  found  satisfy  the  equation  of  the 
dy 

locus.     K,  at  any  point  thus  determined,  y  has  but  one  real  value  for  the  particular 
value  of  X,  and  on  both  sides  of  it  y  has  two  or  more  real  values,  this  point  is  a 

multiple  point.     Its  species  can  be  determined,  as  before,  by  evaluating  -^-  =  -,  for 

the  particular  values  of  x  and  y  which  locate  the  point. 

Ex.  1.  Test  for  multiple  points  y  =  {x  —  a)  vx  +  b. 

SoiiUTiON.  — Since  \/x  is  both  -{-  and  — ,  y  has  in  general  two  values.  But  it  is 
evident  that  for  x  =  0,y  has  but  one  value,  namely, 
6  ;  also  for  x  =  a,  y  has  but  one  value,  6.  These 
are  the  critical  values  of  x  and  y.  Upon  the  point 
(0,  h),  we  observe  that  the  branches  do  not  pass 
through  it ;  since  for  x  negative  y  is  imaginarj^ 
Hence  (0,  b)  is  not  a  multiple  point.  But  upon 
the  point  (a,  b)  we  observe  that  y  has  two  real 
values  on  each  side  of  it.      This  is  therefore  a 

-■     ,  n         .   i.      XT       ^?/             Sx  —  a       ,.,    f  Fig.  126. 

double  pomt.     Now  --  =  ± —,  which  for  ^^^-  -^^"• 

(^  2k/x 

x  =  a  gives  —  =:  ±  \/a.     .• .  The  point  is  of  the  first  species,  and  the  tangents 
dx 


SINGULAR  POINTS.  123 

to  the  curve  at  the  point  make  angles  with  the  axis  of  x  whose  tangents  are 
-f-  \/o„  and  —  s/a.     The  form  of  the  curve  is  given  in  the  figure. 

Ex.  2.  Examine  y^  ==  x^  —  x'^  for  multiple  points. 

Ex.  3.  Examine  x'^  -{-  2ax^y  —  ay^=0  for  multiple  points. 

Solution. — As  it  is  not  easy  to  discover  by  inspection  all  the  points  to  be  examined 

du 

dv  dx 

m  this  case,  we  will  proceed  by  the  second  method.     We  find  --  =  ~  -^  =  — 

dx  du 

dy 

4-05^  ■  I  -  4^0,^0/ 

- — — ^^.    Whence  ix^  -j-  ^(^^V  =  0 ,  and  2ax^  —  3ay^  =  0.     These  equations  give 

the  following  critical  values  \      \  ^  ;     1  ^     ^    \    and  i  f 

(j/  =  0'(2/  =  —  la  \y  =z  —  la. 

But  of  these  only  the  first  set  satisfy  the  equation  of  the  curve.  The  point  (0,  0) 
is,  therefore,  to  be  examined.  Since  hone  but  even  powers  of  x  are  involved,  a 
change  in  its  sign  does  not  change  the  form  of  the  function  ;  hence  the  form  of 
curve  is  the  same  on  both  sides  of  the  axis  of  y.  As  the  equation  is  a  cubic,  there 
it  at  least  one  real  root,  and  hence  one  branch  at  least  passes  through  the  origin 
in  the  plane  of  the  axes.  To  determine  whether  the  other  roots  are  real  or  imagi- 
nary, and  hence  whether  the  other  branches  lie  in  the  same  plane  with  the  axes 
we  might  solve  the  equation.     But  this  is  not  necessary.     We  can  more  readily 

determine  the  facts  by  examining  the  tangents.     Evaluating  --  = '- 11^ '^— 

^  dx  2«x-^  —  3«2/2 

for  a;  =  0,  y  =  0,  we  find  :^-  =  0,  +  \/2  and  —  \/2.     Therefore  there  are  three 

tangents,  and  the  point  is  a  triple  point  of  the  first  species.  The  curve  is  that 
given  in  Fxq.  124,  i^l82). 

Ex.  4.  Examine  ay^  —  x'^y  —  ax"^  =  0  for  multiple  points. 

du  du 

Sug's.  —The  values  arising  from  —  =  0,  and  —  =  0,  are  a;  =  0,  w  =  0,  and  x  = 

dx  dy  '  ;/         ' 

a  ^S,  y  =  —  a.     But  only  the  first  satisfy  the  equation  of  the  curve.     Evaluating 

dy      3x-'y-{-3ax-^      0  .     ^.  .  ^    -,  /     .  .     dv\ 

dx  =   3at-x-  =  0  ^^"  *^"'"  ^^^^^^'  ^"  ^^^  ("^^^^  ^  ^^"  dx>  P^=  1>  P^  - 1  =^  0, 

or  (p  —  l)(p2  -L  p  -f-  1)  =  0.  Whence  p  r=  1,  or  —  ^  dr  \\/~3.  Hence  we  see 
that  there  is  but  one  tangent  in  this  plane,  and  therefore  but  one  branch  passing 
through  the  origin,  and  no  multiple  point. 

Ex.  5.  Show  x^  -{-  .cc2?/2  —  ^ax'^y  -f  a^y"^  =  0  has  a  multiple  point  of 
the  second  species  at  the  origin. 


124 


PEOPEBTIES   OF   PLANE   LOCI. 


CUSPS. 

IS 4:.  Def. — A  Cusp  is  a  variety  of  tlie 
second  species  of  double  point,  in  whicli 
the  osculating  branches  terminate  in  the 
point.  Cusps  are  of  two  kinds  :  1st,  When 
the  branches  lie  on  different  sides  of  the 
tangent ;  2nd,  When  the  branches  He  on 
the  same  side  of  the  tangent. 


18  S,  JPvoh, —  To  examine  a  curve  for  cu^s. 


Fig.  12X. 


Solution. — The  process  is  tlie  same  as  for  multiple  points  of  the  second  species, 
the  only  difference  being  that  the  branches  stop  at  the  point  instead  of  running 
through  it ;  and  hence  that  the  values  of  y  are  real  on  one  side  and  imaginary  on 
the  other. 

To  ascertain  of  which  kind  the  cusp  is,  we  may  compare  the  ordinates  of  the 

curve  in  the  vicinity  of  the  point,  with  the  corresponding  ordinate  of  the  tangent ; 

<Py 
or,  by  means  of  -~,  ascertain  the  direction  of  curvature  ;  or  we  may  construct 

the  curve  about  the  point.  By  the  first  method  we  discover  that  the  cusp  is  of  the 
''^first  kind,  if  the  ordinate  of  the  tangent  is  intermediate  in  value  between  the  cor- 
responding ordinates  of  the  curve  ;  and  that  it  is  of  the  second  kind,  if  the  ordi- 
nate of  the  tangent  is  less  or  greater  than  both  the  corresponding  ordinates  of  the 
curve. 
If  the  common  tangent  is  perpendicular  to  the  axis  of  x,  it  is  best  to  discuss 

the  cusp  with  respect  to  the  axis  of  y,  using  --,  etc. 

Ex.  1.  Examine  (?/  —  h)^  =  (x  —  ay  for  cusps. 


Solution. — ^We  have  y  =  h  ±  {x  —  a)^,  from 

which  we  see  by  inspection  that  for  x:=:a,  y  has 

but  one  value,  for  x  <;  a,  y  is  imaginary,  and  for 

x^a,  y  has  two  real  values.      Therefore  (a,  b) 

1 

,2 


is  the  point  to  be  examined. 


■  ay 


Y 

P 

< 

* 

■"-s^ 

T 

A 

Ct.     I 

D        E 

:                    X 

Fig.  128. 


which  for  x  =  a  becomes  zh  0.     Hence  the  two 

tangents   are   seen   to   coincide,    their   common 

equation  being  y  =  'b;   and  there  is  a  cusp.     To  determine  the  kind  of  cusp,  we 

consider  the  values  of  the  ordinates  of  the  curve  for  x  a  httle  greater  than  a,  as 


a  -\-  h,  h  being  an  infinitesimal.     Substituting,  we  have  y  =  'b:h{a-\-h  —  a)    = 
h  zizh^.    Thus  we  see  that  one  of  the  ordinates  of  the  curve,  asSE=&-|-A  ,!>&, 

3. 

the  corresponding  ordinate  of  the  tangent ;  and  the  other, as  S'E=?)  —  h  ,<^b. 
The  cusp  IS  therefore  of  the  first  kind. 


8INGULAR  POINTS. — CONJUGATE  POraTS. 


125 


Ex.  2.  Show  that  y  =  a  -{-  a;  -{•  bx^  -{-  cx-^  has  a  cusp  of  the  second 

5. 

kind,  if  the  sign  of  ^^  be  considered  as  ambiguous,  and  that  the 
equation  of  the  tangent  at  the  cusp  is  y  =  x  -]-  a. 

SuG. — To  determine  tlie  kind  of  cusp,  we  liave  -r-^  = 

1 
26  d=  ht-cx^,  both  of  whicli  values  are  -\-  for  infinitesimal 

positive  values  of  x.      Therefore  both  branches  of  the 

curve  are  convex  downward  in  the  vicinity  of  the  point, 

and  the  cusp  is  of  the  second  kind.     The  curve  has  the 

general  form  represented  in  the  figure.     There  is  a  point 


of  inflexion  in  the  lower  branch  at  £C  = 


&2 


225c2' 


,  and  it  cuts 


Fig.  129. 


the  tangent  at  ic  ==  — . 


Ex.  3.  Show  that  cy"^  =  x^  has  a  cusp  of  the  first  kind  at  the  origin. 

Ex.  4  Show  that  (y  - —  b  —  cx^y  :=:  (^  —  a)^  has  a  cusp  of  the 
second  kind  at  (a,  6  -f-  c^"^)* 


CONJUGATE  POINTS. 

186,  Def. — A  Conjugate  JPoint  is  an  isolated  point  the  co- 
ordinates of  which  satisfy  the  equation,  while  in  the  vicinity  of  the 
point,  and  on  each  side,  real  values  of  one  co-ordinate  give  imaginary 
values  to  the  other. 

III.— In  the  equation  y  =  {a  -\-  x)\/x,  if  x  is  nega- 
tive, y  is,  in  general,  imaginary  ;  but  for  the  particular 
value  ic  =  —  a,  y  =  0.  Hence  P  is  a  point  in  the  lo- 
cus ;  and  as  there  are  no  other  points  in  this  plane 

adjacent  to  it,  P  is  an  isolated  or  conjugate  point.     On         P       A  "\  X 

the  right  of  the  origin  any  real  value  of  x  gives  two 
real,  numerically  equal  values  to  y,  with  opposite  signs. 
The  curve  has  therefore  two  infinite  branches  on  this 
side,  which  are  symmetrical  with  respect  to  the  axis  -p^      -ioq 

of  X. 

187*  JPvop, — At  a  conjugate  point  some  one  or  more  of  the  differ- 

^'    1  rr     ■        ^      ^1      ^^J     ^^1     ^*Y        .  .       . 

entiat  coefficients  — ,  -^,  -= — ,  —-,  etc.,  is  imaginary. 

Dem. — ^Let  y  =f{x)  be  the  equation  of  a  curve  having  a  conjugate  point  at 
(x,  y).  Then  letting  h  represent  an  infinitesimal  increment  or  decrement  of  x,  and 
y'  the  corresponding  value  of  y,  we  have  y'=^f(x  ±  li)  ==  an  imagmary  quantity,  from 
the  definition  (186).     But 

h'2  (Py       h^ 


y  =A^ 


^       ^  —  da;  1  ^  ck'^   I  .  2 


da;3  1  .  2  .  3 


-]-,  etc. 


126  PROPERTIES  OF  PLANE  LOCI. 

Now  as  y  and  h  are  both  real,  to  make  y'  imaginary,  some  one  or  more  of  the 

„  .     ,    dv  d-y  d?y     ,  ,     .        . 

coefacients  ^,  -^,  -7—,  etc.,  must  be  miaginary.     q.  e.  d. 


18 S,    J*VOp. — Let     <p(x,  y)  =  u  =  0    be    the     equation    of    a 
curve,  freed  from   radicals;    if  there  is  a  conjugate  point  at  (x,  y), 

du  du 

the  partial    differential    coefficients    —  and  —   are    each    equal   to   0, 

du 

,    dy  dx        0 

and    —  =  —  --  =  -. 

dx  du        0 

Dem. — Let  -T^^  be  the  first  differential  coefficient  which  is  imaginary  {187)- 

Take  the  nth  derived  equation  of  u  =  q)[x,  y)  =  0,  and  we  have  (see  Calculus  112  , 

du  d"y   ,  d"u  .         .  ,     ,  .      , 

T-  T^H h  ^—  =  0,  m  which  the  omitted  terms  are  made  up  of 

dy  dx"  '  da;"  ^ 

differential  coefficients  of  u  with  respect  to  x  and  y,  and  differential  coefficients  of 

2/  with  respect  to  x,  of  lower  orders  than  the  nth.     Now,  the  former  are  rational, 

since  u  =  q){x,  y)  does  not  contain  radicals,  and  differentiating  does  not  introduce 

them  ;  and  the  latter  are  rational  by  hypothesis.     Hence,  in  order  that  the  first 

ctut 
member  of  the  derived  equation  may  be  0  (which  is  a  rational  quantity),  —  must 

=  0  and  thus   destroy  the  imaginary  factor  •— ^.       Again,    —-}-—.  y-  =  0 

du 

(Cal.     112  )  ;  whence  as  -r-  =  0,  --  =  0,  and  --  = t  =7^-     Q-  e.  d. 

^  '  dy  dx  da  du      Q 

dy 


ISO,  JPvoh, — To  examine  a  curve  for  conjugate  points. 

du  du 

Solution. — Since  at  a  conjugate  point  --  =  0,  and  --  =:  0,  if  we  find  the  values 

of  X  and  y  which  satisfy  these  equations,  these  values  make  known  the  points  to 
be  examined  ;  i.  e.,  they  are  the  critical  values,  the  same  as  in  the  case  of  multiple 

points.     Having  determined  the  critical  values,  we  may  form  -^,  — ,  — ,  etc.  ; 
^  ^  '  -^  dx  dx2'  dx» 

and,  if  for  the  co-ordinates  of  any  point  under  consideration,  any  one  of  these 

coefficients  becomes  imaginary,  that  point  is  a  conjugate  point. 


ScH. — The  labor  of  producing  the  higher  orders  of  differential  coefficients  is 

cly 

often  so  great,  that  it  is  better,  if  —  does  not  become  imaginary,  to  examine 

dx 

the  point  by  substituting  successively  a-^h  and  a  —  h  f  or  ^  in  the  equation 


SINGULAR   POINTS.— SHOOTING  POINTS.  127 

of  the  curve,  a  being  the  value  of  x  to  be  tested,  and  h  an  infinitesimal. 
If  both  values  of  y  found  in  this  way  are  imaginaiy,  the  point  is  a  conju- 
gate point. 

Ex.  1.  Examine  ay'^ — a;3-f  4aa;2 — 5a2^  +  2a3=0  for  conjugate  points. 

Solution  .    -v-  =  —  ^^  +  8ax  —  5a2  =  0,  and  --  =  lay  =  0,  give  x  =  a,  v  =  0, 
dx  ay 

and  x=^a,  y  =  0.     Only  the  first  two  of  these  values  (a,  0)  satisfy  the  equation  of 

du 

dy           dx 
the  curve  ;  hence  this  point  is  to  be  examined.     To  do  this  we  form  --  = r-  = 

dy 

— .     To  evaluate  this  for  a;  =  a,  y  =  0,  we  nave 


'lay 


dy        6xdx  —  Sadx        —  la  dx  ,. 

-}■  =  T—. =  -n —  -r^  ioxx  =  a,y  =  0. 

dx  'lady  la    dy 


Whence  -r^=  —  1,  or— =\/— 1.     As  this  is  an  imaginary  quantity,  x  =  a, 
dx^  dx 

2/  =  0  is  a  conjugate  point. 

Ex.  2.  Examine  y^  =  x{x  +  aY  for  conjugate  points. 

There  is  a  conjugate  point  at  x  =  —  a,  y  =  0. 

Ex.  3.   Examine  x"*  —  ax^y  —  axy^  +  a^y^  =  0  for  conjugate  points. 

There  is  a  conjugate  point  at  (0,  0). 

Ex.  4.  Examine  {c^y  —  x^y^=  {x  —  a)^{x  —  6)6for  conjugate  points, 
a  being  greater  than  b. 

Sug's.  — There  is  a  conjugate  point  at  x  =  6,  2/  =  — •     Neither  ^,  nor  — ^  are  im-  \ 

d^y      I 
aginary  for  these  values,  though  ^—  is. ,    The  better  way  to  solve  this,  is  to  find 

dx^     / 

a^  63 

the  critical  values  x=^a,  y  =  —,  and  a;  =  6,  y  =  — ,  as  usual.      Then  substituting 

in  the  equation  of  the  curve,  we  find  that  both  points  satisfy  the  equation,  and 

hence  are  to  be  examined.      Then  substitute  in  the  equation,  solved  for  y,  the 

values  a  -\-  h  and  a  —  h.     These  give  real  values  for  y  on  one  side  of  the  point  and 

a^ 
imaginary  values  on  the  other.     Hence  x==a,  y==-;-is  not  a  conjugate  point.     In 

the  same  way  substitute  in  the  value  of  y,  b  zL  h,  and  y  is  found  to  be  imaginary 
on  both  sides  of  the  point. 


SHOOTING  POINTS. 

190,  Def. — A  Shooting  J^oint  is  a  point  at  which  two  or 
more  branches  of  a  curve  terminate,  while  each  branch  has  a  differ- 
ent tangent  at  the  point. 

[Note.  — This  subject  is  not  of  sufficient  importance  to  justify  an  extended  discussion.    We  shall 
tnerely  give  a  couple  of  examples.] 


128 


PROPERTIES    OF    PLANE    LOCI. 


Ex.  1.  To  show  that  y  =  x  tan-^  -  =  x  cot-i  x,  has  a  shooting  point 
at  the  origin,  if  we  limit  the  discussion  to  cot"^  x  numerically  <  Jt. 

SoiiUTioN.— For  a;  =  0,  we  have  ?/  =  0  •  cot— ^0 
=  0  •  sTT  =  0  ;  hence  the  curve  has  a  point  in  the 
origin.  In  the  vicinity  of  the  origin,  i.  e.,  for 
very  small  values  of  x,  x  and  cot— ^ic  have  the 
same  sign,  both  being  +  on  the  right  of  the 
origin,  and  both  —  on  the  left ;  therefore  2/  is  -|- 
near  the  origin  and  the  curve  lies  above  the  axis. 
d^y  2 


Moreover,    -^  =  — 


wherefore     the 


Fig.  131. 


dic2  (1  +  x^y^ 

branches  on  both  sides  of  the  origin  are  concave  towards  the  axis  of  x,  and  there 
is  a  salient  point  at  the  origin,  as  in  the  figure.  To  show  that  there  are  two  tan- 
gents to  the  curve  at  this  point,  and  hence  that  it  is  not  a  cusp,  to  which  it  bears 


some  resemblance,  we  form  -zr  =  cot— ^x  —  - — ; — -. 

dx  1  -\-  x^ 

and  for  x  =  —  0,  is  —  ^Tt. 


This  for  x  =  -{-  0  is  -\-  iit; 


X 


Ex.  2.  To  show  that  y  = j  has  a  shooting  point  at  (0,  0). 

1  +e^ 


Sug's. — For  X  small  and  -}-,  y  is  -]-  5   ^^^  ^OJ^  ^ 
small  and  — ,  2/  is  — ;  hence  the  branches  lie  as  in 

1 

= 1 

dx 


the  figure.     Agam  —  = -j- 


1  +  e^ 


x{l  +  e^y 


-—,  which 


dy 


for  ic  =  4~  0,  gives  -?  =  0  ;  and  for  x  =  —  0,  gives 
dx 

dv 

-^  =  1.     Therefore  there  is  a  shooting  point  at  (0,  0). 


Fig.  132. 


STOP   POINTS. 

191»  Def. — A  Stoj}  J^oint  is  a  point  at  which  a  single  branch 
of  a  curve  terminates. 

Ex.  1.  To  show  that  y  =  x  log  a;  has  a  stop  point  at  the  origin. 

Solution. — In  this  curve  for  all  -f-  values  of  x  less  than  1,  y  is  —  and  has  but 
one  value  ;  for  x  =  1,  y  =  0;  and  for  x^l,  y  is  -\-  and  has  but  one  value.  There 
is  therefore  a  single  branch  of  the  curve- *exten ding  to  the  right  from  the  origin, 
below  the  axis  of  x  from  .r  =  0  to  a;  =  1,  and  above  the  axis  of  x,  beyond  x  =  1. 
But  for  X  negative,  y  is  impossible.     .  • .  There  is  a  stop  point  at  the  origin. 


Ex.  2.  Show  that  y  =  e  ^  has  a  stop  point  at  the  origin. 


TBACING   CURVES.  129 

SECTION  V. 
Tracing  Curves, 

192,  Def. — Tracing  a  Curve  is  discovering  from  the  equa- 
tion of  the  curve  and  its  derived  functions  the  general  form  and  lead- 
ing pecuharities  of  the  curve,  and  its  position  with  reference  to  the 
assumed  axes,  so  that  the  mind  can  conceive  the  locus,  or  that  it  may 
be  sketched  without  going  through  the  details  of  substituting  a  series 
of  values,  as  was  done  in  Section  II.,  Chapter  I. 

ScH. — While  it  is  practicable  to  give  certain  general  directions  for  tracing 
curves,  mucli  has  to  be  left  to  the  ingenuity  of  the  student,  as  the 
infinite  variety  of  forms  of  equations  renders  different  methods  expedient 
in  different  cases.  Nor  do  we  know  how  to  trace  the  loci  represented  by 
every  form  of  equation :  this  would  be  equivalent  to  solving  equations  of 
all  degrees. 


193.  JProh, — To  trace  a  plane  curve  given  by  its  equation  referred 
to  rectangular  axes. 

Method  of  Solution.— If  practicable,  put  the  equation  in  the  iorm.  y  =:  f{x). 
Notice  where  it  cuts  the  axes.  Observe  the  Hmits  and  infinite  branches.  Examine 
infinite  branches  for  asymptotes.  Find  the  direction  of  curvature  between  estab- 
lished or  characteristic  points.  Determine  positions  of  maxima  and  minima  ordi- 
nates.  Sometimes  it  may  be  serviceable  to  ascertain  the  direction  of  the  curve  at 
certain  points,  as  where  it  cuts  the  axes,  by  means  of  its  tangent  at  those  points. 
Notice  the  position  and  character  of  singular  points. 

ScH. — In  giving  the  above  method  of  tracing  curves,  it  is  not  meant  that 
the  processes  there  detailed  are  necessarily  to  be  gone  through  with  in  the 
order  given,  nor  in  fact  that  they  are  all  to  be  applied  in  tracing  the  same 
curve.  These  are  only  means  to  be  used  as  occasion  may  require.  Again, 
while  these  processes  are  general,  and  constitute  what  is  usually  called 
"tracing  curves,"  there  are  other  methods  better  adapted  to  certain  cases. 
Of  these  we  shall  give,  in  the  sequel,  three  ;  viz.,  one  when  the  equation  can 
be  put  into  the  form  y  =  cp[x)  =fc  •^(.r),  in  which  y  =  cp[x)  is  a  diametral  locus 
to  that  represented  by  the  entire  equation ;  another  by  transformation 
from  one  set  of  rectilinear  co-ordinates  to  another  ;  and  a  third  by  passing 
from  one  system  of  co-ordinates  to  another,  as  from  rectilinear  to  polar. 
But  we  will  first  attend  to  a  few  examples  by  the  general  method. 

Ex.  1.  Trace  the  curve  y^  =  ax^  -{-  hx^. 

Solution.— We  have  y  =  ±  xVa  -\-  hx.     On  this  we  observe  that  for  x  =  0,\ 


130 


PROPERTIES   OF   PLANE   LOCI. 


y  =  0.      .• .  Tke  curve  passes  tlu-ough  the  origin.     For  y  =  0,  x  =  0,   or  —  -. 


.  • .  The  curve  cuts  the  axis  of  x  also  at  —  -. 

0 


For  all  n^ative  values  of  x  between  0  and  —  T'  2/  ^^  ^®^^'  ^^^  be^'^ond  —  - 


m  a 


negative  direction  y  is  imaginary.     .  • .  x  ■==  —  -  is  the  limit  of  the  curve  in  this 

direction.     But  for  all  positive  values  of  x,  or  for  aU  values  of  a;>  —  -,  y  has  two 

numerically  equal,  real  values,  affected  with  opposite  signs.  .*.  The  curve  is 
symmetrical  with  respect  to  the  axis  of  x,  and  has  two  infinite  branches  extending 
to  the  right. 


Again  --  =  ±  .'     '    "    — .  which  for  x=  —  y,  becomes 


dx 


s/a  +  hx 


dx 


db  ex,  and  for  ic  =  0, 


dy 
dx 


rh  s/a-     .  * .  At  ( ,  0  )  the  curve  cuts  the  axis  of  x  perpendicularly,  and  at 

(0,  0)  it  cuts  it  in  two  directions,  viz.,  at  tan-i(  -f-  v^a),  and  tan-i(  —  \/a).     This 
also  shows  that  (0,  0)  is  a  multiple  point,  a  double  point. 

Examining  for  direction  of  curvature,  we  have  -^  =  ±: -,  which  is 

^'  4(a  +  hx)^ 

db  between  0  and  —  -,  and  ±  between  0,  and  -\-  cc,     .  • .  At  the  left  of  the  origin, 

the  curve  is  concave  towards  the  axis  of  x,  and  at  the  right,  convex. 

"We  have  a  maximum  and  a  minimum  ordinate  at  a;  =  —  -^^f  y 


36' 


— -v^d«,  as 


ct    1    ^hx 
appears  by  solving  the  equation  ±  — — ^—  =  0. 

s/a  -j-  hx 

It  only  remains  to  examine  the  infinite  branches  for 

asymptotes. 

^  dx  ihx^ 

X  =^x  —  y-r  = 
^dy 


=  CO,  for  re  =  00  ;  and 


a  -)-  f  6x 

_  dy         ^  ihv'^  . 

Y  =  y  —  x-f-  =  — '  =  =P  GO,  for  x  =  oo. 

^^       s/a  4"  bx 
Therefore  there  are  no  asymptotes. 

From  this  investigation  the  curve  is  readily  conceived  to 
have  the  form  given  in  the  figure,  which  is  constructed 
assuming  a  =  36. 


Fig.  133. 


Ex.  2.  Trace  tlie  curve  y^  =  a^x\ 

Results.  The  curve  is  symmetrical  with  respect  to  the  axis  of  x ; 
extends  only  to  the  right ;  is  convex  to  the  axis  of  x ;  has  two 
infinite  branches  ;  has  a  cusp  of  the  first  kind  at  the  origin,  with 
the  axis  of  x  for  the  common  tangent ;  and  has  no  asymptote. 


X 


Ex.  3.  Trace  the  curve  2/  =^  i    ,      ,- 

^        1  -\-  x^ 

Results.  The  curve  cuts  the  axes  at  the  origin  under  an  angle  of  \7t ; 
has  one  infinite  branch  extending  to  the  right  above  the  axis  of  x. 


TRACING   CURVES. 


131 


and  another  extending  to  the  left  below  this  axis  ;  has  a  maxi- 
mum ordinate  at  ^"  =  +  1,  and  a  minimum  at  j;  =  —  1  ;  has  the 
axis  of  X  as  an  asymptote  to  both  branches  ;  has  points  of  in- 
flexion at  (0,  0),  and  at  ^  =  \/3,  and  x  =  —  v  3  ;  between  the 
latter  points  is  concave  towards  the  axis  of  x,  and  beyond  them 
is  convex. 

Ex.  4.  Trace  y^  =:  a^  —  x^. 


Ex.  5.  Trace  {y  —  x'^)"  ==  x\ 

Ex.  6.  Trace  ay^  —  x^  +  bx'^  =  0. 

Results.  The  curve  cuts  the  axis  of  x  at  right 
angles  at  (6,  0)  ;  has  a  conjugate  point 
at  the  origin ;  has  points  of  inflexion  at 
x=  ^h  ',  is  concave  to  the  axis  of  x  from 
x=ib  iox==^b,  and  convex  beyond  ;  has 
two  infinite  symmetrical  branches  with- 
out asymptotes. 

Ex.  7.  Trace  ay^  —  .r^  +  {b  —  c)x-^  +  bcx 
=  0. 

The  form  of  the  curve  is  given  in  the 
figure.  Observe  that  when  c  =  0  this  locus 
becomes  identical  with  the  preceding,  which 
is  sometimes  called  the  campanulate  (bell 
shaped)  parabola. 


Ex.  8.  Trace  the  FoHum  of  Des  Car- 
tes, whose  equation  is  y'^  —  Saxy  -f 
x^  =  0. 

Ex.  9.  Trace  y^  =  2ax'^  —  x^ 


Fig.  134. 


Fig.  135. 


Ex.  10.  Trace  ?/  = 


x^ 


X 


-.     Examine 

a 


the  curve  for  asymptotes,  for  maxima 
and  minima  ordinate s,  for  cusps,  for 
direction  of  curvature,  and  points  of 
inflexion. 


Fig.  136. 


132 


PROPERTIES   OF   PLANE   LOCI. 


•Ex.  11.  Trace  y'  = 


x^  +  x-^ 


Examine 


X  —  1 

the  curve  for  asymptotes,  for  limits,  and 
for  maxima  and  minima  ordinates. 


194,  J*TOb, — To  trace  a  curve  of  the 

second  order,  that  is,  the  locus  of  Ay^  -f- 
Bxy  +  Cx2  +  Dy  +  Ex  +  E  =  0,  by  direct 
inspection  of  its  equation. 

Solution. — One  method  of  solving  this  problem  has  been  given  on  pages  46 — 49. 
The  present  method  is  given  as  a  good  algebraic  exercise,  and  in  illustration  of 
the  remark  in  the  preceding  schoHum  upon  equations  which  take  the  form 
y  =  (p[x)  =t  ipix). 

Solving  the  equation  for  y  we  have 


1st.  K  we  construct  the  straight  line  of  which 

y  =  —  - — (B.X  -[-  D)  is  the  equation  (let  it  be  rep- 

resented  by  M  N  in  the  figure),  any  value  of  x 
(as  AD)  which  locates  a  point  (as  P)  in  this  line, 
locates,  in  general,  two  points  (P',  P")  in  the 
curve,  on  opposite  sides  of  the  line  and  equally 
distant  from  it,  this  distance  being  the  radical  part 


■  4:A  G)x^  +  2(i?i>  —  2AE)x  +  {D^  —  4.AF). 


of  the  value  of  y. 


Therefore  2/  =  ■ —  irii.^^  +  -^)» 


Fig.  138. 


is  a  diameter  of  the  locus. 

2nd.  For  such  value  or  values  of  x  as  render  the 
radical  0,  y  has  but  one  value,  and  at  this  point,  or  these  points,  the  locus  cuts  its 
diameter.     Hence  {B'^  —  4.AG)x-^  +  2(£i)  —  1AE)x  +  (X>^  —  4.AF)  =  0  deter- 

-(JBx  +  X»).     In  general,  this 


2^^ 


mines  where  the  locus  cuts  the  diameter  y  = 

gives  two  values  of  a*,  indicating  that  the  locus  cuts  its  diameter  in  two  points,  as 
in  the  ellipse  and  hyperbola.  But  if  B^  —  4.AC  =  0,  the  equation  becomes 
2{BD  —  2AE)x  -f  (1)2  —  4.AF)  =  0,  which  gives  only  one  point  of  intersection, 
as  in  the  parabola,  a  result  which  agrees  with  the  fact  that  B^—4AG=0  characterizes 
a  parabola  {62).  Locating  the  point,  or  points,  at  which  the  curve  cuts  its  diam- 
eter, we  know,  if  there  are  two  points,  and  the  curve  is  an  ellipse,  that  it  Hes  be- 
tween these  hmits,  or,  if  an  hyperbola,  beyond.  These  facts  will  readily  appear 
by  observing  whether  intermediate  values  of  x  give  real  or  imaginary  values  to  ?/. 
Thus  the  limits  of  the  curve  appear, 

3rd.  If  the  locus  is  an  ellipse,  the  values  of  y  midway  between  the  two  values  of 


TRACING   CURVES. 


133 


X  which  correspond  to  the  extremities  of  the  diameter,  make  known  a  diameter 
parallel  to  tangents  at  the  extremities  of  the  former,  and  hence  determine  the  cir- 
cumscribed parallelogram.     Thus  the  situation  of  the  ellipse  becomes  known. 

dth.  If  the  locus  is  an  hyperbola,  we  can  determine  a  few  values  of  y  corres- 
ponding to  values  of  x  without  the  limits,  and  thus  locate  the  curve.  It  is  often 
expedient  to  find  the  intersections  with  the  axes. 

5th.  If  the  locus  is  a  parabola,  having  determined  its  diameter  and  vertex,  a  few 
values  of  x  will  make  known  sufi&cient  points  to  enable  us  to  sketch  the  curve. 
The  intersections  with  the  axes  may  also  be  of  service. 


Ex.  1.  Trace  the  curve  whose  equation  is 
y2  —  2a;y  +  2^7*  +  2?/  +  a?  +  3  =  0. 

Sug's. — Since  B'^  —  4:AC  <^  0,  the  locus  is  an  el- 
lipse.    Solving  for  y,  we  have 

y  =  OS  —  1  ±  \/ —  .r2  —  Sx  —  2  ; 
whence  y  =  x  —  1  is  a  diameter,  which  we  construct. 

n/ —  x-  —  dx  —  2=0,  gives  x  =  —  1,  and  —  2,  the 
limits  of  the  curve.  Between  these  limits  y  is  real, 
and  without  them  it  is  imaginary.  For  a;  =  —  li, 
2/  :=  —  2,  and  —  3.  Thus  we  find  the  circumscribed 
parallelogram. 


Fig.  139. 


Ex.  2.  Trace  the  curve  whose  equation  is  t/2  -|-  2xy  —  2x^  —  4?/  — 
07  +  10  =  0. 


Sug's. — As  B^  —  iAC  >*  0  the  locus  is  an  hy- 
perbola. ?/  =  — ^  +  2drv/3(x2— iP  —  2).  y  =  —  X 
-f-  2  locates  N  M .  From  3(aj'^  —  x  —  2)  =  0,  we 
find  P  and  P  ",  at  .r  =--  2,  and  —  1.  Between 
these  values  y  is  imaginary  ;  hence  the  locus  lies 
beyond  these  points  to  the  right  and  left.  Put- 
ting 2/  =  0,  we  have  —  2a;2  —  a;  -[-  10  =  0,  whence 
a;  =  2,  and  —21,  and  the  curve  cuts  the  axis  of 
a;  at  C  and  B.  For  ic  =  4,  2/  =  3  •  5  and  —  7-5 
nearly,  and  we  have  1  and  2.  In  hke  manner 
a§  many  points  as  we  wish  may  be  found  ;  but 
with  the  diameter  and  intersections  with  the 
axis,  little  or  nothing  more  is  necessary  in  order 
to  form  a  pretty  definite  idea  of  the  situation  of 
the  curve. 


Fig.  140. 


Ex.  3.  Trace  the  locus  y^  —  2xy  -\-  x^  —  4y  -f-  a;  -f  4  =  0. 


134 


PEOPEETIES   OF  PLANE  LOCI. 


Sug's. — Since  JB^  —  4:AC  =  0,  the  locus  is  a  parab- 
ola, y  =  x  -\-  2  is  the  equation  of  a  diameter.  For 
X  =1  0,  y  Tz=.2.  For  x  negative,  y  is  imaginary.  For 
.r  =  3,  2/  =  8,  and  2. 

Ex's.  4  to  7.    In  like  manner   trace  the  fol- 
lowing :    2/2  _|_  ^xy  +  3j^2  —  4^  =  0  ;   ?/«  — 
^xy  +  2j72  —  2^  =  0  ;    y^  _|_  4^1/  _|_  4j;2  —    y 
4=0;    and  2/"  —  ^^y  +  2^2  _j_  2i/  —  2^  + 
3  =  0. 


Fig.  141. 


lOS,  I*VOh, — To  trace  a  locus  of  the  second  order  hy  means  of 
transformation  of  co-ordinates. 

Solution. — "We  will  illustrate  this  method  by  an  example.  The  method  itself  is 
altogether  too  tedious  for  practical  purposes,  but  is  highly  important  as  giving  a 
clear  view  of  a  process  which  we  have  occasion  to  use  for  other  purposes.  Let  us 
trace  the  locus  whose  equation  is  y^  -\-  2xy  -{-  Sx^  —  4:X  =  0. 

This  is  an  ellipse,  since  B'^  —  4:A  C'<^  0.  We  will  find  its  equation  when  referred 
to  its  own  axes.  This  requires  transformation  from  one  rectangular  system  to 
another.  The  formulce  for  this  transformation  are  a;  =  x,^  cos  a  —  yi  sin  a  -\-  m, 
and  y  z=z  Xi  sin  a  -\-  yi  cos  a  -\-  n.     Substituting  these  in  the  equation,  we  have 


=0. 


(Eq.  A.) 
As  the  required  form  of  the  equation  is  Ay'^  -\-  Bx-  -\-  F=  0,  we  desire  to  elimi- 
nate the  terms  containing  x^y^,  and  y^  and  ccj.  To  find  the  direction  of  the  new 
axes,  I.  e.  to  determine  the  value  of  a,  and  to  find  the  position  of  the  new  origin, 
i.e.,  to  determine  the  values  of  m  and  n,  which  will  effect  this  reduction,  we  place 
the  coefficients  of  the  terms  to  be  eliminated  each  equal  to  0,  and  solve  the  result- 
ing equations.     These  equations  are 

(1)  2  sin  a  cos  a  —  2  sin2  a  -f-  2  cos"^  a.  —  6  sin  a  cos  a:  =  0  ; 

(2)  2n  cos  a  -\-  2m  cos  a  —  2n  sin  a  —  6m  sin  a  -}-  4:  sin  a  =  0  \ 


cos'^a 

?/r  4-2  sin  a  cos  a 

Viyi-\-    sin^a 

.•Ci2-f-27icosa 

?/i-}-2nsin  a 

Xi-\-  n^ 

■2  sin  a  cos  a 

—          2  sin -a 

-f-2sinacosa 

4-27ncosa 

-f  2msin<a: 

-t-2mn 

4-  3  sin2a 

-\-          2cos2a 

+        3cos2a 

— 2n  sin  a 

4-2?i  cosa 

-f3m2 

— 6  sin  a  cos  a 

— 6msin  a 
-f-  4sina 

-j-6mcosa 
—  4  cos  a: 

—  4.7n 

(3)  2n  sin  a  -\-~  2m  sin  a  -\-  2n  cos  a  -\-  Qm  cos  a 

From  (1)  we  find  sin  a  =  .92388,  and  cos  a  =  - 

_,  r.  1  ■  m  -\-  n 

From  (2)  we  have  tan  a  = 


or  —  2.414  = 


4  cos  a  =  0. 
38268,  whence  a 
m.  -\-  n 


112°  30'. 
and  from 


(3),  2.414  = 


n  +  37^  —  2 


71+ 3m  — 2'   "*        ~'  "       n-)-3m  — 2 
Solving  these  equations  we  find  m  =  1,  and  ?i  =  —  1 


m  -f-  71 

Substituting  these  values  of  sin  a. ,  cos  a,  m,  and  n  in  (Eq.  A. ),  we  have,  after 
reduction, 

3.4142/i2  +  .5858a;i2  —  2  =  0, 

as  the  equation  of  the  ellipse  referred  to  its  own  axes.     This  gives  the  axes  as 
3.7,  and  1.53. 


TEACING   CURVES. 


135 


To  locate  the  curve  we  have  but  to  construct 
the  new  origin  at  (1,  —  1)  as  Ai,  and  drawing 
AiXi  making  an  angle  of  112°  30'  with  the  prim- 
itive axis  of  X,  make  A  i  Y  i  perpendicular  to  it, 
and  on  these  axes  construct  an  ellipse  whose  axes 
are  3.7,  and  1.53. 

Ex.  Trace  by  means  of  transformation 
of  co-ordinates  the  locus  whose  equation 
isx2  —  6^t/  +  2/2  —  6jp4-2?/  +  5=0. 

Results.  The  new  origin  (the  centre  of 
the  hyperbola)  is  at  (0,  —  1).  The 
transverse  axis,  which  is  the  new  axis  of  x  makes  an  angle  of 
135°  with  the  primitive  ;  and  the  transformed  equation  is  2^/2  — 
4072  —  4  =  0. 


A 

h 

A 

^Y 

^^         X. 

Vi 

r^ 

Fig.  142. 


196,  J*Tob, — To  trace  a  Polar  curve. 

Method  of  Solution.  ^ — 1st.  Assign  such  values  to  0  as  give  easily  determined 
values  of  r  :  these  will  usually  be  such  as  0,  lit,  it,  \7C,  lit,  etc.  ;  or,  if  some  mul- 
tiple of  G  is  involved  in  the  equation,  like  parts  of  these  values.  Thus  if  sin  29 
is  involved,  making  0  ^  0°,  sin  20  =  0  ;  if  0  =  15°,  sin  20  =  i  ;  if  0  =  45°, 
sin  20  — - 1,  etc.  Construct  these  points.  This  will  often  be  sufficient  to  determine 
the  locus. 

2nd.  Form  —  and  observe  when  (for  what  values  of  r  and  0)  r  and  0  are  mcreas- 


d0 


dr 


ing  functions  of  each  other  and  when  decreasing.     When  ^  =  0  the  point  is  an 

apsis,  X.  e.  one  at  which  the  curve  is  at  right  angles  to  the  radius  vector  :  at  such 
a  point  T  is  a  maximum  or  minimum.  Thus  in  the  elhpse  when  the  pole  is  taken 
at  the  focus  the  vertices  of  the  transverse  axis  are  apsides. 

3rd.  Examine  the  curve  for  asymptotes,  direction  of  curvature,  points  of  inflexion, 
and  any  other  peculiarities  which  may  be  suggested  at  this  stage  of  the  proceeding. 


Ex.  1.  Trace  the  lituus  ^  =  -];• 


Solution. — The  unit  angle 
being  that  whose  arc  equals  ra- 
dius is  about  570.3.  Now  let- 
ting «  =  1,  and  0  =  1,  2,  3,  4, 
5,  and  6,  successively,  we  get 
r  =  ±  1,  ±  .7,  ±  .58,  rb  .5, 
zb  .45,  ±  .41,  zb  .4,  etc.  Lo- 
cating the  positive  values,  we 
get  the  points  1,  2,  3,  etc. ;  and 

locating  the  negative  values  we  have  —1,  —2,  —3,  etc.    The  two  branches  are 
symmetrically  equal. 


Fig.  143. 


136 


PROPERTIES  OF  PLANE  LOCI. 


Again  —  = ,  a  being  =  1 ;  whence  it  appears  that  r  and  9  are  decreasing 

functions  of  each  other  throughout  all  their  values,  and  the  curve  makes  an  infinite 
number  of  revolutions  around  the  pole,  commencing  from  oo  when  6  =  0,  and 


reaching  the  pole  when  6  ■=  oo.     70  = 


*»3 

2" 


0,  gives  r  r=  0.     The  curve  cuts  the 


radius  vector  obhquely,  being  parallel  to  it  at  00,  and  approaching  perpendicular- 
ity as  r  approaches  0,  or  6  approaches  00.     The  pole  is  an  apsis. 


Since  for  6  =  0,   r  =  00,  the  subtangent 


r2c?6 


-,  is  0  for  6  ^=  0,  and  the 


dr  r 

polar  axis  is  an  asymptote. 

To  discuss  the  direction  of  curvature,  we  obtain  the  equation  of  the  spiral  in 
terms  of  the  perpendicular  from  the  pole  upon  the  tangent.     This  equation  is 

p  = ;  whence  -j-  = 5 .     There  is  a  point  of  inflexion  at  r  =  rb  \/2, 

(r^+4)'"  ^''       (7-^-f-4V' 

0  =:  2,  B  and  B'.     From  B  to  the  right  this  branch  is  convex  toward  the  pole  ; 
and  from  B  toward  the  left  it  is  concave,  as  appears  from  considering  the  sign  of 

-f  ,  for  r  >  \/2,  and  for  r  <i  n/2. 
dr 

Ex.  2.  Trace  the  locus  whose  polar  equation  is  r  =  a  sin  3^. 

Solution.— If  6  =  0°,  30°, 
60°,  90°,  120°,  150O,  I8O0,  suc- 
cessively, r  =  0,  a,  0,  — a,  0, 
a,  0. 

dr 

—  =  3a  cos  30,  which  is  positive 
dQ 

from  0  =  0,  to  0  =  BOO,  negative 
from  0  =  30°  to  0  =  90°,  positive 
from  S  =  900  to  0  =  150°,  etc. 
"Whence  we  see  that  r  begins  at 
0  when  0  =  0°,  increases  tOl 
0  =  30°,  diminishes  as  6  passes 
from  30°  to  60°,  becomes  0°  for 
6  =  60°,  continues  to  dimin- 
ish (becoming  negative)  as  0 
passes  to  90^,  becomes  — a,  at 
90°,  etc,  [The  pupil  should  trace  r  through  an  entire  revolution,  in  both  positive 
and  negative  directions.] 

dr 

—  =3a  cos  30  =  0,  gives  apsides  at  0  =  30°,  90°,  and  150°,  L  e.  at  B,  C,  and 
dd 

D.  in  the  figure. 

As  r  never  ==  oc,  there  is  no  asymptote. 


The  equation  in   terms  of  the  perpendicular  upon  the  pole  is  p  = 


■f.2 


(9a-'— Sr^)^ 


,            dp         18a  r  —  Sr' 
whence  — -  = and  the  curve  is  always  concave  toward  the  pole. 


dr 


{9a^  —  8rO« 


RATE  OF  CURVATURE. 


137 


Ex.  3.   Construct  the  locus  whose  equation  is  x^ 
by  first  passing  to  the  polar  equation. 

Solution. — The  polar  equation  with  the  pole 

sin  6  .         .  „  /^^ 

at  the  ongin  is  r  =  a -(cos^Q  —  sin2  9). 

cos4  6 

From  6  =  0°  to  0  =  45°  r  is  real,  finite  and 

passes  from  0  to  0.     Therefore  there  is  a  loop 

dr 


ax'^y  +  o.y^  ==  0, 


in  the  first  octant. 
1  —  3  sin2  6  —  2  sin^  0 


Letting  a  =  1,   -  = 
=  0,  gives  a  maximum 


M        T       M' 
Fig.  145. 


radius  vector  for  0  z^  32°  nearly,  r  =  .45. 

From  0  =  45°  to  0  ^  135°,  r  is  negative.     We  will  first  examine  the  values  of  1 

between  0  =  45°  and  90°.     This  gives  a  continuous  curve  in  the  6th  octant,  A3M. 

Putting  the  equation  in  the  form  r  =  sec  0 (tan  0    -  tan^  0),  we  observe  that  as 

•  tan  0  >>  1  from  0  =  45°  to  0  =  90°,  r  rapidly  increases,  and  becomes  —  go  at  0  =  90°. 

The  branch  in  this  octant  is,  therefore,  infinite.    To  ascertain  more  fully  the  char- 


acter of  this  branch,  we  form  the  subtangent    Subt  z=  —-— 

dr 


rHQ      sin20(cos20  —  sin20)2 


X 


COS" 


tan20  sec  0(1  —  2  sin-^0)2 


cos'^0 
Now  since  between  45°  and  90°, 


1  —  3  sin20  —  2  sin-*0  1—3  sin20  —  2  sin40 

sin20  is  between  ^  and  1,  tan  0  between  1  and  oo,  Bec20  between  2  and  oo,  and  tan  0 
and  sec  0  increase  much  more  rapidly  than  sin  0,  it  is  easy  to  see  that  subt.  con- 
stantly increases  and  becomes  —  oo  at  0  =  90°.  .  • .  This  is  a  parabolic  branch 
and  approaches  to  parallelism  with  A"1". 

Finally,  since  r  :=/(sin  0,  cos  0),  and  only  even  powers  of  cos  0  are  involved,  the 
values  of  r  will  be  repeated  in  the  inverse  order  as  0  passes  from  90°  to  180°. 

x^  +  x^ 


Ex.  4.  Trace  the  locus  whose  equation  is  y'^ 

to  the  polar  equation. 

SuG. — The  polar  equation  with  the  pole  at  the  origin  is  r 
(See  Ex,  11,  193,) 


X 


-,  by  passing 


cos  0(1  —  2  cos^  0) 


■^♦»- 


SUCTION'  VL 

Eate  of  Curvature. 

197 •  ®EF. — The  Curvature  of  a  plane  curve  is  its  rate  of 
deviation  from  a  tangent,  and  is  measured  by  the  subtenses  of  indefi- 
nitely small  but  equal  arcs. 

III. — Let  M  N  and  mn  be  any  two  circles,  AT"  and  AT"'  tangents,  and  AS 
and  AS'  infinitely  small  but  equal  arcs.     Then  will  TS  and  T'S',  drawn  per- 


188 


PROPERTIES   OF  PLANE  LOCI. 


pendicular  to  the  tangents,  be  the  subtenses  which  measure  the  curvature  of  the 
arcs  of  the  respective  circles  ;  and  we  shall  have  curvature  o/"  M  N  :  curvature  of 
mn  : :  T"S  :  "T'S'.  That  curve  is  said  to  have  the  greatest  curvature  which  de- 
viates most  rapidly  from  its  tangent ;  thus,  in  circles, 
the  greater  the  radius  the  less  the  curvature  ;  i.  e.,  the 
curvature  and  radius  are  inverse  functions  of  each 
other.  It  is  also  evident  that  the  circumference  of 
the  same  circle  has  the  same  curvature  at  all  points  ; 
while  in  other  curves,  as  the  conic  sections,  the  curva- 
ture varies  at  every  successive  point.  In  the  ellipse 
the  cui-vature  varies  from  its  maximum  at  the  extrem- 
ities of  the  transverse  axis  to  its  minimum  at  the  ex- 
tremities of  the  conjugate  axis.  In  the  parabola  and  Fig.  146. 
hyperbola  the  curvature  is  greatest  at  the  vertex  and  diminishes  as  the  point 
recedes,  becoming  0  at  infinity. 

It  is  the  object  of  this  section  to  present  a  method  of  measuring  curvature,  and 
of  comparing  the  rates  of  curvature  of  the  same  curve  at  different  points,  and  to 
ascertain  the  law  of  variation.  For  this  purpose  a  circle  is  used,  called  the  oscu- 
latory  circle. 

198.  Def. — An  Osculatory  Circle  is  a  circle  which  has  the 
same  curvature  as  a  given  curve  at  a  given  point  ;  or,  it  may  be  de- 
fined as  the  circle  which  has  the  closest  contact  with  a  given  curve  at 
a  given  point. 


III. — ^Let  BDEC  be  an 
ellipse.  K  with  the  centres 
upon  DC  various  circumfer- 
ences be  passed  through  D,  it 
is  evident  that  they  will  coin- 
cide in  very  different  degrees 
with  the  elUpse.  Some  will 
fall  within,  and  others  without. 
Now  the  one  which  coincides 
most  nearly,  as  in  this  case 
M  N,  is  the  osculatory  circle 
of  the  ellipse  at  the  point  D. 
The  arc  of  the  osculatory  cir- 
cle in  this  case  is  exterior  to 
the    ellipse.      The   osculatory 

circle  at  the  vertex,    as  m"n"  is    within,  and  at  any  other  point,  as  P,  cuts  the 
ellipse,  as  will  be  shown  hereafter. 

199,  T>E¥.—The  Madiits  of  Curvature  is  the  radius  of  the 
osculatory  circle ;  The  Centre  of  Curvature  is  the  centre  of 
the  osculatory  circle  ;  and  the  point  of  closest  contact  is  the  point  of 
osculation. 


BATE  OF  CURVATUBE. 


139 


200,  T>EF.— Contact.  Let  M  N 
and  M'N'  be  two  curves  whose  equa- 
tions are  respectively  y  =f(^x)  and 
y'  =  (p(^x').  Suppose  the  curves  to 
have  a  common  point  P,  so  that  for 
X  =  x' :^= /KD,y  =  y' =  PD-  Now 
if  X  and  x'  take  the  infinitesimal  in- 
crement D  D ',  which  we  will  represent 
by  h,   designating   the    corresponding  Fig  148. 

values  of  y  and  y',  by  Y  and  Y'  (S  D'  and  S'D')?  we  have 

dy        d'^y  h^      d^y     h^        d^y        h" 

dx        c/ip2   2      ^  2 


Y 

'         ^ 

M/^ 

l^ 

A 

D        D' 

X 

Y=/(^  +  /i)=?/  +  ^/i+t1  -^  + 


In?        d^y 
"~3  "^^^ 


2-3-4 


and  X'=^(p{x^K)  =  y'-\-  ^fi 


d^y'h^     d^y'   h^ 


diy'     /i< 


+,  etc. ; 


-f ,  etc. 


dx'"  '  dx'^1   '  dxJ^2-Z  '  ^j7'''2-8-4 
Subtracting  the  second  of  these  equations  from  the  first,  we  have 

Y  -  T  =  (3,  -  y)  +  (j^-^>  +  ( J-^.)-2  +  (Js-A^jan +•  *• 
Now  the  contact  of  these  curves  will  evidently  be  closer  as  Y  —  Y' 
(SS')  is  less.  We  may  therefore  notice  the  following  degrees  of  con- 
formity : 

1st.  If  in  the  case  of  any  two  loci  whose  equations  y  =f(^x)  and 
y'  =  (p(^x'),  there  is  no  value  of  x  =  x'  which  renders  y  =  y',  there  is 
no  common  point. 

2nd.  If  for  x  =  x',  y  =  y'  and  the  differential  co-efficients  are  un- 
equal, the  contact  is  the  slightest,  and  is  mere  Intersection, 

du       du 
3rd.  If  in  addition  to  y  =  y',  -^  =  ~,  and  the  succeeding  coeffi- 
cients are  unequal,  the  contact  is  closer  than  before  and  is  called 
Tangency.     This  is  called  contact  of  the  First  Order. 

dy^dy'  ^^^d^y^d^ 
^ '  dx       dx"  dx^       dx'''^' 

and  the  succeeding  coefficients  unequal,  the  contact  is  of  the  Second 
Order,  etc. 

201*  ScH. — A  geometrical  elucidation  of  this" subject  is  obtained  by  con- 
sidering that  "an  infinitesimal  element  of  the  curve  commencing  from  a 
given  point,  being  straight,  is  coincident  with  the  tangent  line  at  that  point ; 
and  the  next  element  of  the  curve,  being  inclined  at  an  angle  to  the  former 
one,  deviates  from  the  tangent.  Now  let  the  two  consecutive  elements 
be  of  equal  lengths,  and  from  the  extremity  of  the  second  let  a  perpen- 
dicular be  drawn  to  the  tangent :  as  this  perpendicular  is  longer  or  shorter, 
the  curve  will  deviate  more  or  less  from  the  tangent,  that  is,  b©  more  or  less 


4th.  If  we  have  at  the  same  time  y 


140  PKOPERTIES  OF  PLANE  LOCI. 

bent."*  Again,  in  general  a  rectilinear  tangent  is  considered  as  having  hoo 
points  in  common  with  a  curve,  and  a  circle  three,  since  through  the  three 
consecutive  points  one  circumference,  and  only  one,  can  be  passed. 

202,  Def. — A  JParameter^  as  the  term  is  used  in  this  and 
similar  discussions,  is  an  arbitrary  constant  entering  into  an  equation 
of  a  locus,  but  which  is  made  variable  by  hypothesis.  Thus  in  the 
equation  y  =  ax  ~{-  b,  a  and  b  are  constants  as  ordinarily  considered, 
that  is  have  the  same  values  throughout  the  same  discussion.  Again, 
they  are  arbitrary  constants,  since  they  may  have  any  values.  Finally, 
we  may  consider  how  a  straight  line  changes  position  when  a  and  h 
vary  continuously.     In  this  case  a  and  h  are  called  parameters. 


20S*  JPvop, — If  one  curve  be  given  in  sjjecies,  magnitude,  and 
position,  that  is  entirely  given,  and  a  second  given  only  in  species,  in  gen- 
eral the  highest  order  of  contact  possible  is  equal  to  the  number  of  para- 
meters in  the  equation  of  the  second  curve  less  one. 

III. — As  this  proposition  usually  seems  to  the  learner  quite  abstract,  we  will 
give  a  familiar  illustration  of  its  meaning  before  proceeding  to  its  demonstration. 
Let  dy'^  -\-  4ic2  =  36  be  the  first  locus.  The  species  is  ellipse ;  the  magnitude  is 
determined  by  the  value  of  the  axes  6  and  4  :  the  form  of  the  equation  determines 
the  position  of  the  locus.  Thus  this  curve  is  given  in  species,  magnitude  and 
position,  or  entirely  given.     Constructing  it  we  have  the  elhpse  in  the  figure. 

Let  the  second  equation  be  that  of  a  circle  in  its  general  form, 
viz. ,  {X  —  m)"  -\-{y  —  w)2  =  r-,  in  which  m,  n,  and  r,  are  arbitrary 
constants,  which  we  propose  to  treat  as  variables,  thus  making 
them  parameters.  It  is  evident  that  the  closeness  of  contact  of 
these  two  curves  will  depend  on  two  things,  the  value  of  the  ra- 
dius, and  the  position  of  the  centre  ;  but  the  position  of  the 
centre  depends  upon  the  values  of  m  and  n.     Hence  the  closeness  ^^' 

of  contact  depends  upon  the  values  of  the  three  parameters  in,  n,  and  r.  Thus  if 
P  be  the  common  point,  by  locating  the  centre  at  C,  and  using  CP  as  radius,  it 
is  evident  that  the  contact  is  much  closer  than  when  C  is  the  centre  and  CP  the 
radius.  There  is  therefore  some  position  of  the  centre  and  some  value  of  the  radius 
which  will  give  the  circle  closer  contact  than  any  other.  Moreover  it  is  evident 
that  we  have  given  the  widest  possible  opportunity  for  varying  the  contact,  by  taking 
that  form  of  the  equation  of  the  circle  which  has  the  three  parameters  m,  n,  andr. 
We  will  now  give  the  demonstration. 

X>EM. — Let  y  ^=f(x),  and  y'  =  (p{x')  be  the  equations  of  the  loci.  In  order  that 
we  may  make  y  =^y'  for  some  value  of  x  =  x  we  must  have  hberty  to  impose  one 
arbitrary  condition  (i.  e.,  to  vary  the  second  locus  in  at  least  one  respect),  but  this 
requires  one  parameter.  If,  in  addition  to  this  parameter,  there  is  a  second  (i  e.,  if 
we  can  vary  the  curve  in  another  respect)  we  can  impose  another  arbitrary  condi- 

*  Price's  Infinitesimal  Calculus. 


KATE   OF   CUKVATURE.  141 

dv      dv' 
tion,   as   y  =  -^,  and  so  on  for  any  number  of  parameters.     Hence  we  see  that 

CvvC  \AJvb 

one  parameter  makes  intersection  possible  ;  two  make  tangency  or  contact  of  the 
first  order  possible  ;  three  contact  of  the  second  order,  etc. 

204:,  Cor.  1. —  The  right  line  can  ham  in  general  no  higher  order  of 
contact  than  the  first  {tangency),  since  its  equation  y  =  ax  +  b  has  but 
two  parameters  2i  andh. 

205 •  Con.  2. — As  the  equation  of  the  circle  in  its  general  form  has 
but  three  parameters,  it  can  in  general  have  no  higher  order  of  contact 
than  the  second, 

200*  Cor.  3. — The  parabola  can  have  contact  of  the  third  order,  and 

the  ellipse  and  hyperbola  of  the  fourth. 

207*  ScH. — This  discussion  assumes  that  y  =f[x),  which  is  given  in  all 
respects,  is  of  such  a  character  as  to  allow  of  any  degree  of  contact.  Of 
course  the  possibilities  of  contact  are  limited  as  much  by  one  of  the  loci  as 
by  the  other.  Thus,  if  the  first  locus  were  a  circle  and  the  second  an 
ellipse,  the  contact  could  not  in  general  be  above  the  second  order,  although 
the  ellipse  has  a  possible  contact  of  the  fourth  order  with  other  curves. 
Again,  in  this  discussion  we  have  said  "in  general,"  since  exceptions  occur 
at  certain  singular  points.  Some  of  these  will  be  noticed  hereafter.  Thus 
far  we  have  given  the  broader  view  of  osculation,  although  for  the  practical 
purpose  of  the  measurement  of  curvature  we  might  limit  our  view  to  the 
circle,  as  we  shall  do  in  the  following  propositions. 


208,  JProb, — To  produce  the  general  differential  formulcB  for  the 
value  of  radius  of  curvature  and  the  co-ordinates  of  the  centre  of  curva- 
ture of  any  plane  curve,  in  terms  of  the  co-ordinates  of  the  given  curve. 

Solution  1. — Let  y=fi^x)  be  the  equation  of  the  given  locus,  and  (x'  —  w)2  -f 
(y'  —  n)2  =  r2  the  equation  of  the  circle.  Now  as  the  equation  of  the  circle  con- 
tains three  arbitrary  constants,  m,  n,  and  r,  we  may  impose  three  conditions  and 
find  the  values  of  these  constants  which  fulfill  them.     The  conditions  requisite  for 

the  closest  contact  which  a  circle  can  have,  are,  tor  x  =^  x',  y  =  v',  —  =  — ,  and 

'^       ^    dx       dx 

d^v       d^v' 

■T^  =  -j^—^.     These  therefore  are  the  conditions  to  be  imposed,  and  from  which 

the  values  of  m,  n,  and  r  are  to  be  obtained.     In  any  given  case  it  will  be  suffi- 

dt/  d'^v 

cient  to  find  the  values  of  y,  j-,  and  j—,  in  the  equation  of  the  locus,  and  also 

ax  ctX'^ 

dv'  d^v' 

the  values  of  y',  —,  and  — ^-,  in  the  general  equation  of  the  circle,  and  equating 

the  corresponding  values  find  from  the  three  equations  thus  formed  the  values  of 
m,  n  and  r. 

But  for  practical  purposes,  general  formulcB  are  more  convenient.  These  are 
readily  produced,  as  follows  : 


142 


PROPERTIES  OF  PLANE  LOCI. 


Differentiating  the  equation  of  the  circle  twice  in  succession  we  have 

(1)  (a;'-m)  +  (2/'-n)^  =  0 

In  these  equations  and  the  general  equation  of  the  circle 

(3)  {X'  —  m)2  -j-  {y'  —  n)2  =  r% 


dy 


d-y 


we  can  now  substitute  the  values  of  y,  -p,  and  ^-^  as  obtained  from  the  equation 


dx^ 


of  the  given  locus  considering  x  =  x',  and  have 


(4) 
(5) 


{X  —  m)  +  (2/  —  w)--  =  0, 


i  +  S  +  c^'— ^^.  =  «' 


(6)  {x  -^  my  -\-{y  —  nY  =  r'. 

In  order  to  solve  ttiese  equations  for  r,  m,  and  n,  we  get  from  (5) 
dy^ 


(7) 


(8)    a;  —  m  = 


(9)     r  =  ± 


o+i:y 


cZx2 


which  substituted  in  (4)  gives 


Substituting  these  values  in  (6)  and  reducing  we  have 


which  is  the  formula  for  radius  of  curvature. 


The  co-ordinates  of  the  centre  (m  and  n)  are  written  at  once  from  (8)  and  (7). 
They  are  , 


(10)     m  =  a;  — 


(11)     n  =  y  + 


(l  +  ^tW 


dx^/dx 

<Py 


,  and 


l  +  ^^ 

d*y 

dx-^ 


Q.  E.  D. 


Solution  2.— Let  P,  P',  and  P",  Fig.  150,  be 
three  consecutive  points  through  which  the 
curve  M  N ,  whose  equation  is  y  =  f{x),  and 
the  osculatory  circle  mn  whose  equation  is 
(x'  —  m)2  -{-  {y'  —  nY  =  r\  pass,  and  between 
which  they  coincide.  PP' ,  and  P'P"  are  then 
to  be  considered  straight  lines.  Pass  a  circum- 
ference through  these  three  points  by  erecting 
perpendiculars  at  the  middle  points  of  the 
chords  PP',  and  PP'.  These  perpendiculars, 
C  D  and  C  D',  are  consecutive  normals.    Hmc& 


Fig.  150. 


KATE    OF    CURVATURE. 


143 


the  centre  of  the  osculatory  circle  may  he  conceived  as  the  intersection  of  two  consec- 
utive normals. 

Having  premised  the  above  fact,  let 
M  N,  Fig.  151  be  a  curve  whose  equation 
is  y=f[x).  Let  PC  and  PC  be  two  con- 
secutive normals.  Then  is  C  the  centre 
of  osculation,  and  PC  =  r,  the  radius  of 
curvature.  Again,  let  s  represent  the 
length  of  the  curve,  and  tp,  the  angle  at  the 
centre  of  the  osculatory  circle  to  radius 
unity.  As  P  and  P'  are  consecutive  points, 
PP'  =  ds,  P L  =  (ix,  P'  L  =  dy,  and  ce  = 
dip  are  contemporaneous  infinitesimal  elements  of  s,  x,  y,  and  tp  respectively. 
Moreover,  drawing  EH  parallel  to  P'C,  the  angle  PHE  =  PCP'  =d^is  the 
corresponding  infinitesimal  element,  of  the  angle  which  the  normal  makes  with 

the  axis  of  ic  ;  or  d^  =  dtan— ^f ^  ),  as  tan 

\      dy  / 

dtp  is  an  arc  at  a  unit's  distance  from 


Fig.  151. 


-^  )  is  the  angle  a  normal 
dy  / 


makes  with  the  axis  of  x.     Now  since  ce 
the  centre,  and  PP'  =  ds,  is  the  corresponding  arc  at  r  from  the  centre,  we  have 

ds 


df 


the  —  sign   signifying   that  s  is  a   decreasing 


(1)     ds  =  —  rdip,  or  r  = 
function  of  tp. 

But  ds  =  ±:  \/dy^  -j-  dx'^ ;    and  differentiating  tan— ^( -^  j  with  respect  to  oo, 

d-y  dx 
at  =  d  tan-.(-  ^) 


dy' 


d-y  dx 


1  + 


dx-        dy-  -\-  dx'^ 


dx 
dy 

Substituting   these  values  of 


ds  and  dip  in  (1),  we  have  r  =  ± 


d^ydx 


dy^xi 


V    ^  dx'^J 


d^ 
dx^ 


,  as  before. 


ds^ 


209 »  ScH. — Since  the  numerator  of  the  value  of  r,  equals  — — ,  and  x  and 

e  are  increasing  functions  of  each  other,  it  is  always  to  be  regarded  as  +  ', 

whence  we  see  that  the  sign  of  r  depends  upon  the  sign  of  — .     There- 

dx'^ 

fore  r  is  to  be  considered  +  when  the  curve  is  convex  downward,  and  — 

when  it  is  convex  upward  [169). 

Ex.  1.  Find  the  radius  of  curvature,  and  tlie  co-ordinates  of  the 
centre  of  osculation  in  the  common  parabola. 


Solution.  —'We  have  —  =  -,  and  t^  =  —  — 
dx      y  dx-^ 


(2/2  4- P'^)' 


dy^\dy 


tSi©  sign.    Again  w  =  a; 


V        dx^Jdx 


pi 


,  neglecting 


1  + 


(2X2 


,  and  n-=^y-\- 


dy^ 
dx^ 


d?y 
(2x2 


V^ 


lU 


PEOPERTIES  OF  PLANE  LOCI. 


210.  Cob.  1. — The  radius  of  curvature  at  the  vertex  of  the  common 
parabola  is  half  the  latus-rectum,,  since  at  this  point  y  =  0,  and  r  = 

i_£_  =  p. 

211,  Cor.  2. — The   radius  of   curvature   in  the   common  parabola 

1 

varies  as  the  cube  of  the  normal,  since  normal  =  (y^  -|-  -p-^)'^,  and  r  = 

(normal)  3 
P^        * 

Ex.  2.  What  is  the  radius  of 
curvature  of  a  parabola  whose 
latus-rectum  is  9,  at  ^  =  3? 
What  are  the  co-ordinates  of 
the  centre  of  curvature  ?  What 
are  they  at  the  vertex?  Con- 
struct such  a  parabola  with  the 
osculatory  circles  in  position. 

Answer.  For  x  =  'S,  r  =^  C  P 
=  16.04;  771=  A  E -=13-1-;  n= 
EC  =  —  6.91.  At  the  vertex 
r=AC'-=4i;  m  =  AC=4^, 
and  n  =  0. 


Fig  152. 


Ex.  3.  Find  the  radius  of  curvature  of  the  ellipse,  and  the  co-ordi- 
nates of  the  centre  of  curvature. 


Suggestions.    --  = 
ax 


-—,  and  -J-  = 


j5»  ,  (AY  H-  B^xT 

— - —  ;   whence  r  =  — - — —-- ; 

A^y-^  A  B^ 


A^Bi  A'i  ^  ^ 


y(AY^-\-B'x-2) 
A^B* 


212,  CoR.  1. — The   radius  of   curvature  at  the   extremities  of  the 
transverse  axis  of  an  ellipse  is  half  the  latus-rectum,,  or  —  /   and  at  the 

...  A' 

extremities  of  the  conjugate  ajois,  it  is  :^. 


213,  CoR.  2. — The  radius  of  curvature  in  an  ellipse  varies  as  the  cube 

(normal)3A2 


of  the  normal,  since  normal  =  —'^■^''J^  +  B-^x^,  giving  r  =  - 


B^ 


BATE    OF    CURVATURE.  145 


Ex.  4.  Find  the  radius  of  curvature  at  ^  =  2, 
and  also  at  the  vertices  of  the  axes  of  the  elHpse 
whose  axes  are  8  and  4.  Find  also  the  centre  of 
curvature,  and  construct  the  osculatrices. 

Results.  At^=:2(P),  (.375,-3.9),  ie.  C,  is  the 
centre  of  curvature  and  r=5.86(PC).  At 
the  vertices  of  the  transverse  axis  C  is  the 
centre  of  curvature  and  r  =  1. 


Fig.  153. 


214*  ScH. — The  centre  of  curvature  being  the  intersection  of  two  con- 
secutive normals,  it  is  always  in  the  normal  drawn  to  the  point  of  osculation. 
Hence  having  found  the  value  of  r  in  any  given  case,  if  we  can  draw  the 
normal  geometrically,  it  is  not  necessary  to  find  the  co-ordinates  of  the 
centre  of  curvature  in  order  to  draw  the  osculatrix.  If,  however,  we  do 
not  know  how  to  draw  the  normal  geometrically,  the  co-ordinates  of  the 
centre  of  curvature  give  a  point  in  it,  whence  it  can  be  drawn. 

Ex.  5.  Find  the  radius  of  curvature  of  logarithmic  curve,  x-=\ogy. 

3 

my 
Ex.  6.  Find  the  radius  of  curvature  in  the  cubical  parabola,  y^  =  a^x. 

3 

6a-*y 

Ex.  7.  Find  the  radius  of  curvature  of  the  curve  y  =  x^  —  ^=  +  1, 
where  it  cuts  the  axis  of  t/,  and  also  at  the  point  of  minimum  ordi- 
nate. How  does  it  appear  from  the  operation  that  the  curve  is  con- 
cave towards  the  axis  of  x  at  the  former  point  and  convex  at  the  lat- 
ter ?     (See  Fig  97.) 

At  the  first  point  r  ==  —  ^  ;  at  the  second  r  ■=  \. 

Ex.  8.  Find  the  radius  of  curvature  of  the  locus  y^  =  Qx^  -f  x^. 
How  does  it  appear  that  this  locus  is  always  concave  towards  the  axis 

ofx?     (See  ^^.106.)  ^_{^^Mj4x^^_ 

'  —  ^x-y 

Ex.  9.  Prove  that  in  the  cycloid  the  radius  of  curvature  equals 
twice  the  normal.  Construct  a  cycloid  and  upon  this  principle  draw 
the  osculatory  circle  at  several  points.  What  is  the  radius  of  curva- 
ture at  the  points  where  the  cycloid  meets  its  base  ?  What  at  the 
vertex  ? 


2 IS.  I^vop, — At  a  point  of  inflexion  a  rectilinear  tangent  to  a  curve 
has  contojct  of  the  second  order. 


146  PBOPEKTIES  OF  PLANE  LOCI. 

Dem. — Let  y  z=f{x)  be  the  equation  of  the  curve,  and  y'  ==  ax'  -\-  h  he  the 

equation  of  a  right  line.     At  a  point  of  tangency  in  general  we  have  for  x  =  x', 

dii       civ'  d^v 

y  =  y',  and  --  =  -^.     But  at  a  point  of  inflexion  y—  =  0.     Also  in  the  equation 

d^v'  d^v       d^v' 

of  the  right  line  -—-  =  0.     .•.  —^  =  -r^,  and  we  have  the  conditions  of  contact 
^  dx'^  dx^       die  2' 

of  the  second  order,     q.  e.  d. 


210*  J^vop, — At  points  of  maximum  and  minimum  curvature  of 
any  plane  curve,  the  osculatory  circle  has  contact  of  the  third  order. 

dr  \         dx'^/ 

Dem. — At  such  points  —  =  0.     Now  differentiating  r  = ,,  we  have 

dx  "^  d'^y 


dx« 
dr 


2V    "^  dx"J    ^    dx\dx-^J        dx\    '^  dx'^J    ^  ^  , 


da;  /^V 

^^          d^y         dxKdx^/ 
Whence  ^  =  . 

^^'  11^ 

"•"daja 

dx^ 
But  in  the  circle  we  have  found  y  —  n  =  —  — -r •     Differentiating  this,  and 

finding  the  value  of  -r^,  we  have  ~  =  ■-."r .     Therefore  as  the  third  differ- 

^  dx^  dx»  1    J   ^ 

"^  dic2 
ential  coefficient  is  the  same  in  the  circle  as  at  a  point  of  maximum  or  minimum 
curvature  of  any  plane  curve,  the  contact  is  of  the  third  order  at  such  points. 

Q.  E.  D. 

217*  Cor. — The  contact  of  the  osculatory  circles  at  the  vertices  of 
the  conic  sections  is  closer  than  at  other  points,  a  fact  which  is  also  appa- 
rent in  the  construction. 


2X8.  JPvop, —  When  contact  is  of  an  even  order  the  loci  intersect; 
hut  ivhen  of  an  odd  order  they  do  not. 

Dem. — Let  Y=f{x)  and  y  =  cp{x)  be  the  equations  of  the  two  loci.  Then  the 
difference  of  their  ordinates  corresponding  to  a;  zh  /?-  is  Y'  —  y'  = 
/dT  d7j\(±h)  upT  cPy\(dzh)^  uUT  d^y\(±h)^  /d*Y  d^iyV^h)^ 
\'dx~dx)  1  '^\dx^~ d^^/~2  '"V'dxS  ~dxV  2Ty+\  di;*  drV2.3.4 
-f-,  etc.  Now,  when  the  order  of  contact  is  even,  the  first  term  of  this  difference 
which  does  not  reduce  to  0,  and  which  fixes  the  sign  of  the  sum  of  the  series, 
contains  an  odd  power  of  ±  /i ;  and  hence  Y'  —  y'  is  positive  for  -j-  /?,  and  nega- 


KATE   OF   CURVATURE.  147 

tive  for  —  h,  showing  that  the  loci  intersect  at  the  point.  If,  on  the  other  hand, 
the  contact  is  of  an  odd  order,  the  first  term  which  does  not  reduce  to  0  contains 
an  even  power  of  zh  h  ;  and  hence  does  not  change  sign  with  h,  and  one  of  the 
curves  lies  within  the  other,  as  in  tangency.     q.  e.  d. 

210»  CoE. — The  osculatory  circle  always  cuts  a  conic  section  except 
at  points  of  maximum  and  minimum  curvature. 


220.  JPvoh, — To  produce  the  formula  for  radius  of  curvature  in 
terms  of  Polar  Co-ordinates. 

Solution. — We  will  produce  this  formula  by  transformation  of  co-ordinates,  as 
the  process  affords  both  a  good  exercise  in  transformation,  and  also  in  changing 
the  independent  variable.  In  order  to  distinguish  between  radius  of  curvature 
and  radius  vector,  let  the  former  be  represented  by  R,  and  the  latter  by  r. 

3. 

We  have  already  seen  that  JB  = — ^  .,    ■■  -  (208).     But  this  formula  was  pro- 

d'^-y  dx 

duced  on  the  assumption  that  x  was  equicrescent,  and  hence  d{dx)  =  0.     To  give 

it  the  more   general  form,  we  have  only  to   remember  that  d^y  =  d(  —  jdx  = 

d^  dx  —  d^x  dy         .  ,  .■,.  ,^  ,  -         ,    . 

— -— ;  and  hence  that  the  general  formula  is 

dx 

3. 

{dx'2  4-  dy-^y 


(1)     E 


d^y  dx  —  d^x  dy 

The  equations  for  transformation  are  y  =  r  sin  0,  and  x  =  r  cos  6.     Considering 
Q  equicrescent  {i.  e.  as  the  independent  variable)  and  differentiating,  we  get 
dy  =  dr  sin  Q  -\-  r  cos  0  dO, 

dy^  ~  dr^  sins  0  +  2r  sin  0  cos  QdrdQ  +  r-'  coss  0  dB\ 
d'iy  =  d^r  sin  0  -f-  2  cos  0  dr  dQ  —  r  sin  0  dQ^, 
dx  =  dr  cos  0  —  r  sin  0  dQ, 

dx'^  =  dr^  cos2  0  —  2r  sin  0  cos  QdrdB  -\-  r^  sins  0  dO^, 
and  d^x  =  d~r  cos  0  —  2  sin  OdrdO  —  r  cos  0  dS^. 

3  3 

.  • .  (dx^-  +  dy^y^  =  (dr^  +  r^de^f,  and 
d^y  dx  —  d^x  dy  =  2(sin20  -\-  cos^Q)dr^dQ  —  rd0(sin20  -f  cos^B)d^-\-  (sinsQ  +  cos20)r2d03 

=  2dr^  dQ  —  rdQ  d^r  +  r^  dB\ 
Whence,  substituting,  we  have 

r-4-r2V 
_  (dr2  4-  r^B'i)''  _       Vd0^  ^     / 

~  MrHB  —  rdSdr  4-  rm^  ~  aIt^         d'-r    ,      "     ^'  ^*  ^* 

dQ-^        dQ^  ^ 

221.    CoE. — Since    the    length   of   a  normal  to   a  polar    curve   is 
_|_  7-2  j 2^  {107),  representing  the  normal  by  N,  we  have 


B=z 


^dr^         d^r 

2 —  r h  r* 


us 


PROPERTIES  OF  PLANE  LOCI. 


Ex.  1.   Find  the  radius  of  curvature  of   the  logarithmic  spiral, 

d 
r  =  a  . 

dr        0,  ^  dV        0,     „ 

Solution.     •—=  a  log  a,  and  — -  :=  a  log^  a. 

2a20  iog2  a  —  TO   log2  a  -f-  r^         a^Q  log2  a  -|-  r^ 
the  polar  normal.     [The  first  reduction  is  made  by  remembering  that  r  =  a  .1 

Ex.  2.  Find  the  radius  of  curvature  of  the  lemniscate  of  BemouiUi, 

r^  =  a^  cos  26. 

dr  a2  sin  20  a  sin  20 

Solution.     -^  = = , 

dQ  r  y  cos  20 


and 


dr^ 
d^r 

dm 


«2  sins  20 
cos  20~' 
2acos2  20  -\-aB\n^2B 

cos^  20 


Substituting  these  values,  we  have 

/a2sin2  20  ,      ^        ^^\| 

( ofl-  +  «^  ^°^  20  y 

\    cos  28  / 


R  = 


2a2sin2  20    ,    2a-2cos2  20  +  a2sin2  20    , 

_| I7777Z h  a2co8  2e 


cos  20 


cos  20 


Vcos  20/ 


«3 


3a2  sin2  20  +  Sa^  cos2  20        3a2^cos  20 

cos  20 


a2 
3r 


^  ♦» 


SECTION   YIL 


Evolutes  and  Involutes. 

222.  Def. — An  Mvollite  of  a  curve  is  the  locus  of  the  centre 
of  curvature.     The  primary  curve  is  called  the  In/colute, 

III. — If  M  N  be  a  plane  curve,  and 
the  centre  of  curvature,  C,  be  deter- 
mined for  any  point,  P  ;  then,  as  P 
passes  along  the  curve  to  P',  P",  P'", 
etc.,  the  centre  of  curvature  will  de- 
scribe another  curve,  as  C ,  C,  C",  C", 
etc.  M '  N  being  thus  described  is  the 
evolutfi  of  M  N  ;  and  M  N  is  the  invo- 
lute of  M'N. 


N  X 


EVOLUTES   AND   INVOLUTES. 


149 


223,  J^rob, — Given  the  equation  of  a  plane  curve,  to  Jind  the  equa- 
tion of  its  evolute. 


m  =  X 


Solution. — Let  y  =f{x)  be  the  equation  of  the  given  curve,  as  M  N  Fig.  154. 
Now  the  co-ordinates  of  the  centre  of  the  osculatory  circle  are  the  co-ordinates  of 
the  evolute.     Hence,  if  we  combine  the  equations 

~'     Idx  ^       ^  ■  ~^  dx^ 

W^'       and    n-=y+~—, 

dx^  dx2 

with  2/  =fix),  and  ehminate  a;  and  y  there  will  result  an  equation  between  ?n  and 
n,  the  co-ordinates  of  the  evolute,  which  is  therefore  its  equation. 

ScH. — The  equation  of  the  locus,  y  =f{x)  is  needed  in  connection  with 
the  values  of  m  and  n,  only  when  these  values  contain  both  x  and  y. 

Ex.  1.  Find  the  equation  of  the  evolute  of  the  common  parabola. 


dv      p  d^v 

Solution. — ^We  have  -f  =-,  and  — 

ax       y  dx^ 


—  — .     Whence 

y3 


m  =  x 


/i  +  ^1!^^ 


d^ 


dx^Mx  __  3y2  +  2p2 
2^       ' 


and  n  =  y  ~\~ 


^  dx^ 
dx^ 


y3 
pi' 

,  if  we  ehm- 


Now  from  m  =  -~r — —,  and  n  = 

inate    y   we    obtain,    after    a    Httle    reduction. 


N    ^N' 


Fig.  155. 


n2  =  27~^^~-P'^"^'  ^^^^^  ^^  *^®  equation  sought.      Tracing  the  curve  we  find 
M'A'N',   Mg.   155.      If  we   transfer  the  origin  to  A',   the   equation  becomes 


n'^  = 


27p 


m^ 


224,  ScH. — This  locus  is  called  the  Semi-cubical  Parabola,  any  curve 
having  infinite  branches  and  no  rectilinear  asymptotes  being  called  a  para- 
bola. 

Ex.  2.  Find  the  evolute  of  the  circle. 


Sug's. — The  equations  to  be  solved  are 


m  =  X —  =  0. 

y2r2y 


r2y3 

y 

y-  r'^ 


0, 


aiid  x^  +  y^  =  r\     Whence  m  ==  0  and  n  =0,  for  all 

values  of  x  and  y,  and  the  evolute  is  a  point,  the  centre.     This  is  evidently  correct, 
since  all  normals  (radii)  of  the  circle  meet  at  the  centre. 


150  PROPEBllES  OF  PLANE  LOCI. 

Ex.  3.  Find  the  evolute  of  the  ellipse. 


Sug's.— "We  have  m  =  ——,  n  =  —  ^,  and  A^y^A-B^x^ 
A^  B^ 

=  A^B\  from  which  to  eliminate  x  and  y  and  find  an 

equation  between  m  and  n,  the  co-ordinates  of  the  evolute. 

The   equation   sought  is  A^rn^  -\-  B  n    =  {A^  —  B'^)  . 
The  evolute  is  of  the  form  CC'C'C",  Mg.  156. 

Ex.  4.    Produce  the  equation  of  the  evolute  to 
the  cycloid. 


dy      \/2ry — ■//- 
dx 


y 


dry 

dx^ 


Stjg's.  — We  have  '^ :=  -■"""'^ — —,  and  ^j—;  = -,.    Whence  m=x-\-2\/ 2ry — y% 


and  n=  —  y.     .- .  y  =  —  n,  and  x  =  m  —  2  v/ —  2ni  —  rv^.     Substituting  these 
the  equation  of  the  cycloid,  we  have  m  =  vers— ^( —  n)-\-\^  —  2rn  —  v!^. 


ui 


22S,  CoR. — The  evolute  of  a  cycloid  is  an  equal  cycloid. 


Yj 

A         E  _D 

( °K) 


Dem. — The  equation 
X  =  vers— ^( —  y)  -\-  V —  2ry  —  y-  is 
the  equation  of  a  cycloid  referred  to 
its  highest  point  A,  as  the  origin  and 
having  a  tangent  at  that  point  as  the 
axis  of  abscissas  and  the  axis  of  the 
cycloid    for    the    axis  of  ordinates. 
This  wiU  readily  appear  by  produc- 
ing the  equation  under  these  conditions.     Thus  in  Fig.  157  AD=AE4-ED  = 
CH  +  FP  =  CB— HB+FP  =  arcHPE  — arc  H  P+ FP  =  arcEP+ FP. 
But    A  D  =  JC,    P  D  =  —  y,   arc  E  P  = 
vers-'EF   =  vers-^PD   ==  ver-^  ( — y\ 


C  H 

Fig.  157. 


and  FP  =  \/EFX  FH  =v/PDX  FH 


=  V\— 2/)02r—  PD)  =  A— y)(,2r+2/)  = 
\/ —  iry  —  y'-.     Hence  x  =  vers— ^( —  y)  -\- 


FiG.  158. 


\/ —  2ry^-—  y^. 

Thus  we  see  that  M  iig.  158  being  a 
cycloid  whose  equation  is  a:  =  vers— ^  y  — 

v/2r2/  —  y-,  N  is  its  evolute  whose  equation  is  m  =  vers-i( —  n)  +  ^ — -^'* —  '^^■> 
referred  to  A  as  its  origin.  This  equation  is  satisfied  for  none  but  negative  values 
of  n,  and  gives  m  =  0,  litr,  4.7tr,  etc.,  for  n  =0  ;  and  also  forn  =  — 2r,  m  =  ;rr, 
3;rr,  etc.,  as  it  should. 


EVOLUTES  AND  INYOLUTES. 


151 


ScH. — The  student  will  readily  dis- 
cern the  character  of  the  evolute  of 
the  cycloid  from  the  property  that  the 
radius  of  curvature  is  always  twice 
the  normal.  Thus  if  the  two  circles 
C,  and  C  roll  along  the  bases  AX 
and  AX'  at  equal  rates  so  as  to  keep 
their  centres  in  the  same  vertical  line 
P'  will  describe  the  evolute  as  P  does 
the  involute. 


5k^ 

•^ 

""^"^ 

/ 

'       \\ 

y 

\ 

A 

V 

^)\' 

X 

A' 


X' 


Fig.  159. 


220,  JPvojy* — A  [produced)  normal  to  an  involute  is  tangent  to  the 
evolute,  the  point  of  tangency  is  the  centre  of  curvature,  and  consequently 
the  normal  thus  produced  is  the  radius  of  curvature. 

Dem.  — Let  (m,  n)  be  any  point  in  the  evolute  of  A IVI, 
from  it  draw  a  normal  to  AM,  and  let  {X,  y)  be  the 
point  at  which  it  is  normal.     The  equation  of  this  nor- 


mal is 


y  —  n 


dx 


or 


dy 
X  —  m  -f-  Y'jy  —  ?2)  =  0  (1) .     Now  as 

the  point  (m,  n)  changes  position  (cc,  y)  also  changes, 
and  to  observe  the  law  of  change  we  differentiate  (1)  for 
X,  y,  m  and  n  as  variables.     This  gives 


Fig.  160. 


dx  —  dm  -{- 


dy-  —  dndy 


dx 


+  {y 


^^df  =  «' 


or 


^+1:+*^ 


d"y        dm 
dx-         dx 


dx 

dndy 

dx^ 


=  0.     (2) 


But  as  (m,  n)  is  in  the  evolute  we  have  {208)  y 
d^y 


d'y 
dx^ 


whence 


^  +  %  +  'y 


n) 


dx-^ 


dm 
dx 


0.     Therefore  dropping  these  terms,  (2)  becomes 
dn  dy 


dx^ 
dx 
dy 


dx 
dy 


dn 

dm 


=  0,  and 

{X  —  m),  which  is  the  equation  of  a  normal  to 


Hence  the  equation  y  —  n  - 

the  involute  at  {x,  y),  may  be  written 

'^Z  -  n)  =  ^(^  -  m\ 
which  is  the  equation  of  a  tangent  to  the  evolute  at  (m,  n).     q.  e.  d. 

227 »  Cor.  —  The  i^adius  of  curvature  and  the  arc  of  the  evolute  vary 
by  equal  increments  ;  that  is,  the  arc  of  the  evolute  between  tivo  centres  of 
curvature  equals  the  difference  between  the  corresponding  i^adii  of  curva- 
ture. 


162 


PBOPERTIES   OF  PLANE  LOCI. 


Dem. — Since  the  radius  of  curvature  is  a  tangent  to 
the  evolute  it  coincides  with  the  arc  between  two  con- 
secutive points.  Thus  P  and  P'  being  consecutive 
points  on  the  involute,  the  radius  at  P  is  to  be  consid- 
ered as  having  the  two  consecutive  points  C  and  C 
common  with  the  evolute  to  which  it  is  tangent ;  and  as 
P  passes  to  P',  the  radius  of  curvature  so  changes  po- 
sition as  to  have  the  consecutive  points  C  and  C" 
common,  and  to  coincide  with  the  curve  between  them. 
Thus  it  appears  that  the  radius  of  curvature  and  the 
arc  of  the  evolute  vary  by  equal  increments. 


Fig.  16L 


228.  ScH. — From  these  relations  it  is  easy  to  see  how  an  involute  may 
be  described  mechanically  from  its  evolute.  For  example,  to  draw  a  para- 
bola, make  a  pattern  of  the  form  AOCM  Fig.  161,  the  edge  OCM  being 
the  arc  of  an  evolute  to  the  required  parabola,  and  AO  =p,  Fasten  a  cord 
at  M  and,  wrapping  it  around  the  edge  of  the  pattern,  fasten  a  pencil  to  the 
free  end  at  A.  Keeping  the  string  tight,  move  the 
pencil  along  as  from  A  to  P,  P',  R,  and  it  will  de- 
scribe the  parabola  which  is  an  involute  to  OM. 
In  like  manner  any  curve  can  be  described  by 
means  of  a  pattern  of  its  involute.  The  cycloid 
and  ellipse  are  drawn  with  special  facility  by  this 
method.     Thus,  for  the  ellipse,  take  a  thin  rectan-      ^  0 

gular   board  ABED,  and  w^on  it  fasten  two  pat-  ^i^-  162. 

terns  ACOD,  and  BC'OE,  the  edges  CO  and  CO  being  the  evolute. 
Then  fastening  at  O,  one  end  of  a  string  whose  length  is  AGO,  the  free 
end  will  describe  the  semi-ellipse  as  it  is  moved  from  A  to  B.  Upon  this 
principle  attempts  have  been  made  to  make  a  pendulum  vibrate  in  the  arc 
of  a  cycloid. 

229,  Cor. — Every  carve  has  one  and  only  one  evolute ;  hut  every 
evolute  has  an  infinite  number  of  involutes,  since  every  point  in  the  string 
describes  an  involute  as  the  string  unwraps  from  the  evolute. 


^»  » 


SECTION  YIIL 

Envelopes  to  Plane  Curves. 

230.  Def. — An  JEnvelope  is  the  locus  of  the  intersection  of 
consecutive  lines,  or  curves,  represented  by  a  given  equation,  when 
one  or  more  of  its  parameters  are  made  variable. 

III.— Let  {x  —  w)2  -j-  j/^  —  r2  —  0  be  the  equation  of  the  locus  whose  envelope 


ENVELOPES  TO  PLANE  CURVES. 


153 


a-b  c  cl 


Fig.  163. 


N' 


is  required.  Let  m  be  the  (variable) 
parameter.  Let  r  =  Bl,  so  that  Baa' 
shall  be  one  position  of  the  given  locus, 
which  in  this  case  is  a  circle.  Now  sup- 
pose m  to  take  an  infinitesimal  incre- 
ment dm,  putting  the  centre  at  2,   and 

giving  {x  —  (m  -f-  dm)}^  +  2/'^  —  r^  =  0 
as  the  equation  of  the  consecutive  locus. 
The  intersections  of  these  loci,  as  a,  a',  are  points  in  the  envelope.  Again,  let  m 
take  another  infinitesimal  increment,  as  2  3,  then  h  and  h'  are  points  in  the  envel- 
ope. In  like  manner  the  intersections  of  3  and  4,  4  and  5,  etc. ,  etc. ,  give  points 
in  the  envelope.  The  envelope  in  this  case  is  evidently  the  two  parallel  right 
lines  MN,  M  N'. 

Were  we  to  make  r  vary  at  the  same  time  as  m,  the  form  of  the  envelope  would 
be  clianged,  and  would  depend  upon  the  relative  rates  of  change  of  r  and  m.  Of 
course,  the  student  will  understand  that  the  points  of  intersection  a,  h,  c,  d,  etc. , 
are  only  in  the  envelope  when  1  2,  2  3,  3  4,  etc.,  are  infinitesimal ;  in  other  words, 
the  envelope  is  the  limit  toward  which  these  consecutive  intersections  approach  as 
the  increments  2  3,  3  4,  etc. ,  diminish. 


231,  JP'TOb, — To  find  the  equation  of  the  envelope  of  a  given  locus. 

Solution.  — Let  F{x,  y,  m)  =0  be  the  equation  of  the  given  locus.  The  consec- 
utive locus  will  be  F^x,  y,  m  -f  dm)  =  0,  or  F{x,  y,  m)  +  d^F^x,  y,  m)  =  0.  If  we 
now  combine  the  equation  of  the  locus  with  this  equation  of  its  consecutive,  elim- 
inating m,  we  shall  determine  the  locus  of  the  intersection,  i.  e.,  the  envelope. 
But  since  Ft^x,  y,  m)  =  0,  the  equation  of  the  consecutive  can  always  be  reduced 
to  d,nF(x,  y,  m)  =  0.  Hence  in  practice  we  simply  combine  the  equation  of  the 
locus  with  its  first  differential  equation  eliminating  the  parameter,  thus  obtaining 
the  envelope,     q.  e.  d. 

ScH. — It  is  of  course  possible  that  the  consecutive  loci  may  not  intersect ; 
as,  for  example,  x'^  -^  y"^  =^  r^,  when  r  is  made  variable. 


Ex.  1.  Find  the  envelope  of  y"  =  m(x  —  m). 


Solution. 
or  0  ==  xdm 


-Differentiating  with  reference  to  m,  we  have  0  =  dni{x  — m) 
-  2mdm.     Whence  w  =  ^x.     Combining  this  with  ;</-  =:  m'x 


as  to  eliminate  m,  i.  e. ,  substituting  Ix  for  m,  we  have  y 
of  the  envelope. 

III. — The  geometrical  significance  of  this  operation 
wiU  be  readily  seen  by  constructing  a  few  parabolas  on 
the  same  axis,  giving  to  m  slightly  differing  values.  The 
consecutive  intersections,  a,  h,  c,  a',  h',  c',  etc.,  will  ev- 
idently approach  the  two  straight  hues  AM,  AM' as 
the  difference  between  the  consecutive  values  of  m  is 
made  less  ;  therefore  these  lines  are  the  envelope  of  the 
parabola  y^  =  m{ps  —  m),  or  the  series  of  consecutive  pa- 
rabolas of  which  four  are  represented  in  the  figure. 


-  mdnit 
m),  so 
^x,  as  the  equation 


Fig.  164. 


154: 


PROPERTIES   OF   PLANE   LOCI. 


Ex.  2.  Find  the  envelope  oi  y  =  ax  -\ ,  a  being  the  parameter. 

Construct  a  figure  illustrating  the  result. 

The  envelope  is  y^  =  4m^. 

Ex.  3.  A  line  of  fixed  length  slides  between  two  fixed  lines  at  right 
angles  to  each  other  ;  required  the  envelope. 

Solution.— Let  the  axes  AX  and  AY  be  tlie  fixed 
lines  at  right  angles  to  each  other,  between  which  the 
line  M  N  of  fixed  length,  as  c,  slides.     Let  A  M  =  b, 

and  A  N  =  a,  whence  tan  M  N  X  = ,  and  the 


equation  of  M  N  is  2/  =  —  -^  +  &,  or  |  +  ^  =  1. 


(1). 


We  have  also  a^-\-b^  =  c'^  (2).     The  most  direct  method 

(not  the  most  expeditious)  would  now  be,  to  find  the 

value  of  a  or  6,  from  (2),  and  substitute  it  in  (1),  which 

would  then  have  but  a  single  parameter  and  its  envelope  could  be  found  as  before. 

But  the  following  method  is  less  tedious  :    Differentiating  (1)  and  (2)  we  have 

ydb    ,^  ^Q  r^y    and  ada  -\-  hdb  =  0  (4). 

b-^   ^    cc^ 
Whence  by  ehminating  da  and  db  between  (3) 

and  (4),  we  have  -  =  -f  ;  which  substituted  m 
^  a         b^ 

(1)   gives  after  reduction    &3  _-  c^?/.       Similarly 
a?  =  c^x.     Substituting  these  values  in  (1),  there 

z  z  2 

results  y"  ■j-x''  —  c\  the  equation  sought. 

III. — This  locus  is  readily  sketched  by  drawing 
M  N  in  slightly  changed  positions,  and  noting  the 
intersections  of  consecutive  lines,  as  in  Fig.  166. 

ScH. — This  locus  is  a  variety  of  Hypocydoid, 
a  kind  of  curve  generated  by  a  point  in  the  cir- 
cumference of  a  circle  rolling  on  the  concave  arc  of 
(within)  a  fixed  circle.  In  this  variety  the  radius 
of  the  fixed  circle  is  4  times  that  of  the  genera- 
trix. 

232,  I*rop, — The  envelope  to  a  plane 
curve  IS  tangent  to  each  of  the  intersecting 
curves  of  the  series. 


Fig.  167. 


Dem.— Let  F[X,  y,  m)  =0  (1),  be  the  given  locus.     But  from  dmF(x,  y,  m)  =  0 
(2),  we  have  m  =  (p(x,  y)  ;    whence    the    equation  of  the   envelope    becomes 

F{x,  y,  cp(x,  y)}  =  0  (2,).     If  now  -^  is  the  same  for  both  the  locus  and  its  en- 


ENVELOPES  TO  PLANE  CURVES. 


155 


velope,  it  follows  that  they  have  a  common  tangent,  wherever  they  have  a  common 
point.     From  (1)  we  get  by  differentiating 

dF(x,  y,  m) 
dF{x,  y,  m) 


dx 


4- 


dF{x,  y,  m) 
dy 


--  =  0,  from  which  -^-  =  — . 

dx  dx  dF{x,  y,  m) 


dy 
Differentiating  (2i)  we  have 

dF{x,  y,  cpjx,  y)}  _^  dF{x,  y,  cpjx,  y)}  ^  dy      dF\x,  y,  (p{x,y)}  ^  /d(p{x,  y) 

dy  dx  dcp{x,  y)  \ 


dx 


+ 


g.  y)\ 

'X      J 


0. 


But  as  by  (2)  dmF{x,  y,  q>{x,  y)]  =  0,  and  q){x,  y)  =  m,  this  becomes  —    '  — \- 

CLvu 


dF{x,  y,  m) 
dy 


dy      .  dy 

-7-,  whence  -— 
dx  dx 


locus. 


dF{x,  y,  m) 

dx 

dF{x,  y,  m) 
~      dy 


',  the  same  as  in  the  equation  of  the 


Ex.  4.  Find  the  locus  to  which  the  hypot-  y 
enuse  of  a  right  angled  triangle  of  con-  ^ 
stant  area  is  always  tangent. 

Solution, — Let  the   constant  area  ABC  =;  a, 


the    parameter    AC   =  m. 


Then    AB    =  --, 
m 


2a 
tan  BCX  = ,  and  the  equation  of   BC  is 

y  = -X  -\ .     The  equation  sought  is  xy  =  -, 

the  equation  of  an  hyperbola. 

Ex.  5.  What  is  the  envelope  of  an  ellipse  which  retains  its  axes  in 
the  same  right  lines,  but  varies  in  eccentricity  so  that  AB  =  a  con- 
stant, 771  ? 

Sug's. — Since  AB  =  m,  the  equation  of  the  ellipse  is  A''y^  -\-  m^x-  =  A"m'^ ;  in 
which  A  is  the  parameter.  The  equation  of  the  envelope  is  xy  =  Im,  an  equilat- 
eral hyperbola  referred  to  its  asymptotes. 

As  will  appear  hereafter,  the  area  of  an  ellipse  is  TtAB.  Hence  the  area  of  the 
above  locus  is  constant. 

Ex.  6.  From  every  point  in  the  circumference  of  a  circle,  pairs  of 
tangents  are  drawn  to  another  circle.  Find  the  locus  to  which  the 
chord  connecting  corresponding  points  of  tangency  is  constantly 
tangent. 

Solution. — Letting  C  be  the  centre  of  the  first  and  A  of  the  second  circle,  it 
is  evident  that  as  P  moves  around  the  circle  P'P"  -will  change  its  position.  The 
envelope  of  P'P"  is  required.  Let  CP  =r,  and  AP"  =  r'.  Let  P  be  designat- 
ed as  (m,  n),  P'  as  (m',  n'),  and  P"  as  {yn",  n"). 

The  equation  of  the  locus    P'P"   whose  envelope  is  i-equired  is  y  -  -  n'  ;= 


156 


PROPERTIES   OF  PLANE  LOCI. 


— ; 77(0;  —  m')    (1).     But  by  reason  of 

the  tangents  PP'  and  PP"  we  have 
nn'  -f-  '^^''n'  =  ^'-  (2)  ;  and  nn"  -f-  inm"  = 
r'-  (3).  Subtracting  (3)  from  (2)  we  have 
n{n'  —  n")  -f-  m{m'  —  m")  =  0  ;  whence 


m  —  m 
m 

= (35- 

n 


—,  and  (1)  becomes  y  —  n' 


m'),  wy  4"  mx  ==  r'2.    Thus  we 


Fig.  169. 


find  the  equation  of  the  given  locus  P'  P"  to  be  ny  -\-  mx  =  r'2  (4). 

Again,  if  we  let  the  distance  between  the  centres  of  the  circles  AC  be  repre- 
sented by  a,  we  have  the  relation  between  n  and  m  in  the  equation 

?i2  4-  (m  —  ay  =  r2     (5). 

The  problem  then  is  to  find  the  envelope  of  (4),  when  the  relation  between  n 
and  m  is  that  given  in  (5).     Differentiating  (4)  and  (5)  considering  m  and  n  as 


variables,  we  have  y— — Ux  =  0, 
dm 


dn  X        ^     dn   ,  .  nx   , 

— —  = ,  and  n- \- m  —  a  =  0.     .*. j- 

am  y  am  y 


m  —  a  =  0,  my  —  nx  =  ay      (6). 

Finally  eliminating  m  and  n  between  (4)  (5),  and  (6),  and  reducing,  we  have 
r-2y'2  _|_  (r2  —  a-);c-  -\-  laf'^x  =  r'^,  as  the  equation  of  the  envelope. 

Hence  the  envelope  is  a  conic  section.     When  a  =  0  it  is  a  circle  ;  when  a  <^r, 
an  ellipse  ;  when  a  =  r  &  parabola  ;  and  when  a  >>  r,  an  hyperbola. 


233,  J^TOb. — An  infinite  number'  of  parallel  right  lines  meet  a 
given  curve  on  the  same  side ;  and  where  each  meets  the  curve  a  line  is 
drawn  making  an  angle  with  the  parallel  which  is  bisected  by  the  normal 
at  that  point.     Required  the  envelope  of  the  line. 

Solution. — Let  M  N  be  the  given 
curve,  PO  one  of  the  parallels,  PQ  the 
normal,  and  PG  the  locus  whose  envel- 
ope is  sought.  Our  first  purpose  is  to 
find  the  equation  of  PG.  Let  y'  =  cp{x') 
be  the  equation  of  M  N ,  and  v  the  tan- 
gent of  the  constant  angle  PSX.  Since 
P,  whose  co-ordinates  are  x',y',  is  a,  point 
in  PG,  the  equation  has  the  form 
y  —  y'  =  a{x  —  x)  in  which  a  is  the  tan- 
gent of  PDX.  Now  PDX  =  PQX  —  DPQ  -- 
—    (PSX      -     PQX)     =    2PQX     —     PSX. 

tan2PQX  —  tan  PSX 
1  -j-  tan  2PQX  tan  PSX' 


IG 

Fig.  170. 
PQX  — QPS  =  PQX 

Therefore    tan  PDX    = 


Again,  as  PQ  is  normal  to  y'  =^  (p(,x'),  tan  PQX 


^^'        -u  .      ot-.^x,  2  tan  PQX 

—  TT-;  ;  whence  tan  2 PQX  = -^ 

dy  '  ^  1  —  tan-^  PQX 


—2  — 

dy' 


1 


dx2 
dy^t 


Substituting,  and 


ENVELOPES  TO  PLANE  CURVES. 


157 


introducing  v  for  tan  PSX,  we  have 

dx'2 
dy'' 


V   —   V 


a  = 


2— 
dy' 


1  —  -T-.- 


dx''-^ 
dy"^ 


dx' 


dx 

2-—V 
dy 


Putting  —7  =  p,  for  convenience,  the  equation  of  PG  becomes 
dy 


y  —  y  = 


p^v 


2p 


[x  —  x'). 


1  —  p'^  —  "Apv 

From  this  equation,  its  first  difierential  equation,  and  the  equation  of  the  curve 
y'  =  ^{x'),  if  x,  and  y'  be  ehminated,  the  resulting  equation  between  a;  and  y 
will  be  the  equation  of  the  envelope  sought.  But  the  difficulties  of  elimination 
are  often  insurmountable.     We  give  two  cases  which  are  readily  solved. 

234,  CoK.  1.—^  O  P  is  parallel  to  AX,  f  =  0,  and  the  equation 
PG  becomes 

2p 


y  — y 


p2  1 


(X  -  X'). 


23S.  Cor.  2. 
Hon  becomes 


-If  OP  is  perpendicular  to  AX,  v=  oo,  and  the  equ^ 


V' 


"        "  2p 

236 »  ScH. — It  is  a  well  known  prop- 
erty of  light  that  its  rays  impinging  upon 
a  reflecting  surface  are  thrown  off  so  as 
to  make  the  angle  between  the  reflected 
ray  and  the  normal,  equal  to  that  between 
the  incident  ray  and  the  normal.  In  con- 
sequence of  this  law,  when  the  rays  of  the 
sun,  which  are  practically  parallel,  are  re- 
flected from  a  curved  surface,  the  inter- 
sections of  the  consecutive  reflected  rays 
produce  a  luminous  curve,  called  a  Caustic^ 
which  is  an  example  of  the  envelope  dis- 
cussed in  the  problem.  The  annexed 
figure  affords  an  illustration.  Let  NAN' 
be  a  section  of  a  circular  cylindrical  mir- 
ror, made  perpendicular  to  its  axis.  Let 
1  to  11  be  rays  of  light  parallel  to  the 
axis  of  the  mirror  AC.  The  envelope  of 
the  reflected  rays  is  the  caustic  curve  NM. 
MN  shows  the  lower  branch  of  the  caus- 
tic, the  rays  not  being  represented.  This 
cnrs'e  may  be  seen  inside  of  a  ring  lying 


(X  -  X'). 


Fio.  171. 


158  PROPERTIES   OF    PLAISE   LOCI. 

on  a  table  in  the  light.  It  is  famihar  to  the  milkman,  as  **the  cow's  foot 
in  the  milk,"  which  is  the  caustic  formed  upon  the  smooth  surface  of  the 
milk  in  a  bright  tin  pail,  by  reflection  of  the  Hght  from  the  inside  of  the 
pail. 

Ex.  1.  To  produce  the  equation  of  the  caustic  when  the  incident 
rays  are  parallel  to  the  axis  of  a  parabolic  reflector. 

Solution.— We  have  y'^  =  4mx',  and  -— ;  =  7--,  4m  being  the  parameter  of  the 


V  ^  =  4mx ,  ana  — -  =  77—, 
^  ay        2m 

4m' 


parabola.     Substituting  this  value  of  p,  and  for  x',  j-,  in  the  equation  {234),  we 


have  after  reduction 

yy'2  —  4m2?/  -\~  Am"y'  =  ^my'x      (1). 

Differentiating  (1)  with  respect  to  the  parameter  y',  gives  y'  = .     Sub- 
stituting this  in  (1),  and  reducing  we  have 

X  =  m  ±  V —  y^,  as  the  equation  of  the  caustic.     This  can  only  be 
satisfied  for  2/  =  0,  x  T=m;  whence  we  see  that  the  caustic  is  a  point,  the  focus. 

Ex.  2.  To  find  the  caustic  to  the  circle  "when  the  incident  rays  are 
parallel  to  the  axis  of  x. 

dx'            11' 
Solution.— Equation  of  circle  y'^ -\- x'^  =  r^.      .*.  ;r-;  = ;•    Equation  of 

etc/  X 

2r)  ,  2x'w' 

reflected  ray  (234:)  y—y'z:^  — —^{^  —  a;'),  becomes  t/  —  y'  = : '—ri^  —  ^)^ 

p-  —  1  2/   —  ^  " 

o^  ^  -  «^'  =-  ^  l~y^  \y'  -  y)'  o^'  ^-^'  =  Kl  ~  ^')^^'~^^  ^^^- 

Differentiating  (1)  with  respect  to  x',  we  have 

y'       ^  y  \,  .        .        m       ^\x' 


—  1 


--:^V-.)-(|,-^,)f.- 


V        £C'2  y"^     }  ^  \x'       y'/y' 

Whence  -2  =  _  (  ^^f  +  l^y^-y)  _1  +  _, 
r'^/'ii'-i  \  x'^ 

-      %  +  '-  vS  +  0^^-  -  y^' 

x"i-\-y'-2         r-Vv'- 4- •'^'^\.  ,         X        -u-  V  J-  -J  J  1.    oi'^ -^  y'^    . 
or  T-^-  =  —-{ — ){y  —  y)  '>   which  divided  by  gives 

X'^  T^  3      3 

—  =  — {y'  —  y).     From  which  we  find  y'  =z  r  y  . 

y  ^     y'^ 

v'  x'  ^ 
To  find  x',  substitute  in  (1)  for  y'  —  y  its  value  '— ;j-,  and  we  have 

?/'2  X'2    2/'x'2  W'2   X'2  ,c        X 

X  —  x'  = = X .     But  X  2  =  r2  —  V  2  whence  x  —  x'  = 

2y'x'        r2  2r^  ^ 

_r2  4-2?/2  ,           ,          2r?x          _,   ,^.        I  i  .       ,    ...    ,                  ,           2r^r 
-X  ,  or  X  = jr-7-.     Putting  r  y    for  2/ ,  this  becomes  x  =  -7 


2^2  r^^2y'^  o     .  .  ^^4.2/ 


BEOTIEICATION  OP  PLANE  CURVES. 


159 


Finally,  squaring  these  values  of  y',  and  x'  and  substituting  in  y"^  -|-  x'^  =  r^,  we 


nave  r  y    -| 


^  —  =  r2,  wMch  is  tlie  equation  of  the  caustic  sought. 


[Note. — At  this  stage  of  the  course  the  student  will  need  to  acquaint  himself  with  the  elements 
of  the  Integral  Calculus,  as  given  in  Chap.  III.   of  the  second  part  of  this  volume.] 


■♦♦»• 


SECTION  IX, 
Rectification  of  Plane  Curves. 

(a)    BY  MEANS  OF  KECTANGULAR  CO-ORDINATES. 

^57.  Def. — To  Rectify  a  Curve  is  to  find  its  length.  The 
term  arises  from  the  conception  that  a  right  line  is  to  be  found  which 
has  the  same  length  as  the  curve. 

238,  I^rop, — The  formula  for  the  rectification  of  plane  curves  is 

dz  =  Vdx.^  +  dy2 ; 

in  which  z  represents  the  length  of  the  curve,  and  x  and  j  the  general  co- 
ordinates. 

Dem. — Let  M  N  be  any  plane  curve,  AD 
=  X,  and  P  D  =:  y  ^®  ^^1  co-ordinates,  and 
let  D  D'  represent  dx  ;  then  will  P'E  repre- 
sent dy,  and  PP' ,  dz  \  i,  e.,  dx,  dy,  and  dz  will 
represent  contemporaneous  infinitesimal  in- 
crements of  the  co-ordinates  and  the  arc.  xC^T 
From  the  right    angled  triangle    PEP'  we     R 


have  at  once     dz  =  \/dx^  -\-  dy^.     q.  e.  d. 


Fig.  172. 


239,  ScH. — To  apply  this  formula  to  any  particular  curve,  we  have 
simply  to  find  dv  or  dy  from  the  equation  of  the  proposed  curve,  substitute 
it  in  the  formula,  and  then  integrate  between  proper  limits. 

Ex.  1.  Rectify  the  semi-cubical  parabola  whose  equation  is  y^==ax^. 
(See  e volute  of  common  parabola.) 

Dem. — Differentiatins;  y-  =  ax^,  we  have  dy  =  -^dx,  whence  dy-  =  -— — dx"^  = 

^a^x^ 
,  '    dx^  =  iaxdx"^.     Substituting  this  value  in  the  formula  for  rectification,  it 
4ax3 

becomes  dz  =  (dx'  ^  ^axdx^)    =(!-}-  l^^)  ^^      Integrating  we  have  z  =* 


160 


PROPERTIES  OF  PLANE  LOCI. 


8  #• 

^;:-(l  -j-  f  aa;)    -}-  C.     To  determine  C  we  may  reckon  the 

length  of  the  curve  from  the  origin  A,  whence  for  cc  =  0, 

8  8 

z  =  0,  and  we  have  0  =  — — \-  G,  or  G=  —  ^r=-.     The  cor- 

A  iCL  A  id 


reded  integral  is  therefore  z  =  ^— — [(1  -\-  ^ax)     —  1]. 

Aid 


To 


illustrate  this  result  consider  A  M  the  curve  whose  length 
is  to  be  found,  or,  which  is  to  be  rectified.  As  this  curve 
is  infinite  in  extent,  we  can  inquire  only  for  the  length  of 
some  specified  portion  of  it.  Let  it  be  required  to  find 
the  length  of  the  curve  between  the  origin  and  the  point 
P  whose  abscissa  we  will  call  h.     Substituting  this  value 


of  X,  we  have  arc  A  P  =  z 


27a 


[(1  +  ^ahy  —  1].     Were 


Fig.  173. 


it  required  to  find  the  length  of  some  other  arc,  as  PP',  we  should  integrate 

between  the  limits  .r  =  A D,  and  ic  =  AD'.     Thus,  let  AD  =  6  and  A D'  =  c. 

8  ^ 

Besuming  the  indefinite  or  general  integral  z  =  ^-{1  +  ^ax)^  -\-  C,   substituting 

A  (CL 

successively  x  =  h,  and  x=  c,  and  subtracting  the  former  result  from  the  latter, 
we  get  for  the  length  of  the  arc  P  P',  the  definite  integral 

Ex.  2.  Bectify  the  common  parabola. 


ifidifi  1  I 

Solution.— From  y"^  =  2px  we  have  dx'^  =  - — ^  ;   whence  dz  =  -( pa  _i_  y^)  (fy. 


To  integrate  this  apply  formula  <(^  of  reduction,  and  we  have 


yVp^  +  y^        p  j        dy 

2p  2^  v/p2  +  y^ 


But 


/: 


dy 


\/p2-|-2/ 


=  =  log[2/+v/p2-h2/^]+C. 


^=^^±^VS-iog[^+v/p-q:F]+c. 


2p 


P^ 


Estimating  the  arc  from  the  vertex,  which  is  the  origin,  we  have  C=  — %^ogp ;  and 

A 


the  corrected  integral  is  z  = 


y\/p'^-\-y^      p,     Vy -\- \/p^ -\- y'^ 


2p 


+|i°gp+^;'+^']. 


ScH. — Instead  of   integrating  as  above,  we  may  expand  (p2  _^  ^^y  by 
Maclaurin's  or  the  Binomial  theorem,  and  then  integrate  each  term  separate- 

1     1     y3       1     1     1     1    3/5       1     1    3     1    1     1     ^7 
ly,obtaimng.=  (y  +  -. -.---.-.-.-.-  +  -.-. --2-. -.-.--etc.) 

+  C. 

Ex.  3.  Bectify  the  circle. 


Solution.     From  x:^  -\-  xf^  =  r^  we  have  dy^  =  —dx^,  hence  dz  =  ( ' —  ) 

y2  \     y     y 


dx 


RECTIFICATION   OF  PLANE   CURVES.  161 

dz  = ' 7,  and  z  =  rsin—^--\-  C,      But   this  is  only  a  restatement  of  the 

s  r 

(r2  —  X") ' 

problem  and  is  of  no  use  in  the  solution.     We  shall  have  to  integrate  in  some 

other  way.     We  may  wi'ite 

-2           /          X'\-i  -^ 

dz  =  r{r'~  —  Q(fi)    da;  =  (  1 ;^  )     dec  =  r(l  —  x^-)  'dx^, 

putting  -  =  Xi  for  convenience.     Expanding  by  Maclaurin's  or  the  Binomial  the- 
orem, and  integrating  each  term  separately,  we  have 

Restoring  x  this  becomes 

ofi 3^5  1  -3  .  5.r7 

2  •  3r^  "^  ii  .  4  .  5ri  "^  2  •  4  •  6  •  Tr- 
To  determine  C,  reckon  the  arc  from  the  axis  of  ordi- 
nates  (B),  whence  for  a;  =  0  z  =  0,  and  therefore  (7=0. 
Then  making  x  =  r  we  have  the  length  of  the  quadrant 


<;  + 


BX,  z  =  r( 


(^  +  2^3+2-:r75  +  2.4.b.7  +  ^'^0 


Fig.  174. 


Bepresenting  the  sum  of  the  series  in  the  parenthesis 
l>y  i"^,  we  have  z  =  a  quadrant  =  irit,  and  the  whole  cir- 
cumference =  27rr.  Letting  D  be  the  diameter,  this  be- 
comes Dit,  whence  it  appears  that  it,  the  sum  of  the  series 
above,  is  the  ratio  of  the  circumference  of  a  circle  to  its 
diameter.  By  extending  the  terms  in  this  series  sufficiently,  reducing  each  to  a 
decimal  fraction  and  adding,  we  find  it  =  3.1415926-f-.  For  practical  purposes  it 
is  usually  taken  as  3,1416,  and  for  still  ruder  approximations  as  3|. 

240,  Cor.  1. — The  circumferences  of  circles  are  to  each  other  as  their 
radii,  or  as  their  diameters. 

24:1,  ScH. — The  quantity  tt  has  not  only  fundamental  importance  in  ge- 
ometry, but  has  great  historic  interest.  Upon  it  depend  both  the  method 
of  obtaining  the  circumference  of  a  circle,  of  a  given  radius,  and  the  area 
of  a  circle,  as  well  as  many  other  problems.  The  ancients  sought  with 
much  diligence  to  discover  its  value.  Archimedes  (287  B.  C.)  found  it  to 
be  between  3^^^  and  3^.  Metius  (1640)  gave  a  nearer  approximation  in  the 
fraction  ff^f.  In  1853  Mr.  Rutherford  presented  to  the  Royal  Society  of 
London  a  computation  by  Mr.  W.  Shanks  of  Houghton-le- Spring,  extend- 
ing the  decimal  to  530  places.     The  following  is  its  value  to  50  places  : 

3.141  592  653  589  793  238  462  643  383  279  502  884  197  169  399  375  10. 

Ex.  4.  Rectify  the  cycloid. 

Solution. — We  have  dx^  =  — ^ — - — ,  whence 

zry  —  2/^ 

fdz  =  (2r)^/(2r  —  y)~'%  =  —  2{2r)\2r  —  y)^  -f  a 
Beckoning  the  arc  from  the  origin   C  =.  4r,  and  the  corrected  integral  is  z  = 


162 


PROPERTIES   OF   PLANE   LOCI. 


.i 


i 


2(2r)''(2r  —  2/)"  +  ^^-    Making  y  =  2r,  2  =  5  the  cycloidal  arc  =  4r,  whence 

the  entire  arc  of  the  cycloid  is  seen  to  be  4  times  the  diameter  of  the  generatrix. 

242,  CoE.  2. — Any  arc  of  a  cycloid,  edimated  franx  ike  vertex,  is 
equal  to  twice  the  corresponding  chord  of  the  generatrix. 

* 

Dem. — Resuming  the  indefinite  inte- 

i  J^ 

gral  z  =  —  2(2r)  ^{2r  —  ?/)-  +  C,  if  we 

estimate  the  arc  from  B,  where  y  =  2r, 

we  have  C  =  0  ;  and  the  corrected  in- 

tegral  is  z  =  —  2(2r)  ~{2r  —  y)~.     This 
is   the  length   of    any  arc   estimated 
from    B,  as  BP,  PD  being  y.     But 
in  the  right  angled    triangle    BEC, 
.  • .  arc  B  P  =  2  times  chord  B  E.     q.  e.  d. 

ScH. — Both  the  fact  that  the  length  of  the  cycloid  equals  4  times  the 
diameter  of  the  generatrix,  and  that  any  arc  estimated  from  the  vertex 
equals  2  times  the  corresponding  chord  of  the  generatrix,  are  readily  ob- 
served from  the  manner  in  which  the  curve  is  described  from  its  evolute. 
Thus  the  radius  of  curvature  at  the  vertex  is  2  times  the  diameter  of  the 
generatrix.  But  this  is  the  arc  of  the  evolute.  So  also  as  the  string  un- 
winds from  the  evolute,  the  radius  of  curvature  is  seen  to  be  the  arc  of  the 
evolute  and  equal  to  twice  the  corresponding  chord  of  the  generatrix. 
(See  Fig.  159.) 

2^2. 

Ex.  5.  Bectify  the  hypocycloid,  whose   equation  is  x'^  -{-  y'^  =  a^. 


BE  =^  v/BC  X   BG   =  v'^TXAr  —  y). 


(See  Fig.  167.) 
Ex.  6.  Bectify  the  ellipse. 


The  length  of  the  entire  curve  is  Qa. 


Solution. — For  this  purpose  the  equation 

2/-  +  (1  —  e^)^:^  =  ^2(1  _  gs) 

is  most  convenient.      From  this  we  have   dy'^  =  —(1  —  e^)^cbfi',  whence 


x^ 
t 


dz  =  ^  \dx^  +  —(1  —  e^-ydx^  = 


dxVy'^  +  x\l  —  e-^)2 
- 


^  ^.^     1(1  —  e^){A^  —  x^)  -Kl  —  e2)2a;2 
(1  —  e2)(^2  —  a;2) 


-  -  -^J- 


—  a;2  4-  x2  —  e2a;2 
A^  —  x^ 


=  dx 


A'^  —  e'^x^ 


Adx 


A^ 


v/^'2 


e^x^ 


1  "■ — li"  )  '  ^® 

Adx       (  ^        e"-x-^  e-'x*  Se^x^ 

have    dz  =  — —  J  1 — 

v^lT^^i  I  2A-2         2.4.^-*        2  .  4  .  6^« 

r ,        ,    /^     dx  e2     /*  x"dx  e*       r  x'^dx  3e^         /•   x"dx 

Jdz  =  A  j- 


—  etc. 


e'±'^\i 
whence 


v/^2 


— ^c^* 


e2     r  x"dx  e*       r  x*dx  2>e^         r 


2.4.6^yy^IZ^, 


RECTIFICATION  OF  PLANE   CURVES. 


163 


—  etc.     Now  integrating  each  of  these  terms  separately  we  have  z  =  ^sin— i • 


3e6 


SbA^Y^A^/A^  .       X       X    ,- \       a;3  /- "l       x^  r-. ) 


2  .  4  .  6^5 
—  etc.,  +  a 

K  we  estimate  the  arc  of  the  ellipse  from  the  extremity  of  the  conjugate  axis, 
we  have  for  a;  =  0,  z  =  0  ;  whence  substituting,  we  find  (7=0. 


It 


j    Again,  making  x  =  A,  and  observing  that  sin— '1  =zs  — ,  we  have 

_7tA     e^/AnA\        e*    r3A^/A  TtAy\ 
*'"~T~2"1V2"T/~2X43LTV2*~2"/J      2- 


3e6      (5A^r3A^/A   TtAyi) 

T6-i^h~LH2-^-JJi-"*^- 

Uniting  and  multiplying  by  4,  we  have  for  the  entire  circumference  of  the  ellipse,  i 
A  o     A/'-i  ^'  ^^'  3.3.5e6  N 

4.,  =  2;r^(l  -^^-  ,.2.4.4  -   2.2.4.4.6.6  "  ''''} 
This  series  converges  more  or  less  rapidly  as  the  eccentricity  is  greater  or  less, 

but  is  always  converging. 


(6)    RECTIFICATION  BY  MEANS  OF  POLAR  CO-ORDINATES 
243  •  JProp. — The  formula  for  rectifying  polar  curves  is 

dz  =  (r2d<?-^  +  dr2)^ 
Dbm. — Let  A  be  the  pole  of  the  curve  MN,  AP  any  radius 
vector,  and  AP'  the  consecutive  position  of  the  radius  vector,  so 
that  PAP'  =  d6,  6  being  the  variable  angle.  Let  z  represent  any 
arc  of  the  curve,  and  r  the  radius  vector,  and  with  A  as  a  centre  and 
radius  AP,  draw  PD.  Then  PP'  =  dz,  and  P'D  =  dr,  are  infin- 
itesimals of  z  and  r  respectively,  and  contemporaneous  with  dO. 
Now  from  the  right  angled  triangle  P  DP,  right  angled  at  D,  we 


►'P  =  \j\ 


Fig.  176. 


have  P'P  =  NJPD  +  P'D  .  Bntdd  being  the  arc  measuring 
PAP'  at  a  unit's  distance  from  A,  PD  =  rd&  ;  whence,  substitut- 
ing dz  for   PP',  dr  for  P'D,  and  rdO  for  PD,  we  have  dz  =  (r^dd-2  -j-  dr^y, 

Q.  E.  D. 

Ex.  1.  Rectify  the  logarithmic  spiral,  log  r  =  0. 


Solution.     dO^  = 
i 


M^dr'i 


.-.  dz  =  {M^  -{.  1)  dr,  and 


z  =  (M^  ^  I)  r  -\-  C.  If  the  arc  be  reckoned  from  the 
pole,  so  that  z  =  0  when  r  =  0,  the  constant  (7=  0,  and  the 

corrected  integral  is  z  =  (M^  +  l)~r.  If  we  take  the  Na- 
pierian logarithm  of  r,  we  have  z  =2  r.     Now  when  0=0 

r  =  1,  and  z  =  2  .  But  by  tracing  this  curve  we  see  that 
if  A  B  Fig.  177,  represent  the  value  of  r(  =  1)  when 
6  =  0,  there  are  an  infinite  number  of  spires  between  this 


Fio.  177. 


164  PBOPERTIES  OF  PLANE  LOCI. 

and  the  pole  (see  110),  and  we  have  the  singular  result  that  their  entire  length  is 
\/2.     When  Q  =  27t  =  6. 2831853-f-,  we  have  log  r  =  6.2831853+,  and  in  the  com- 

mon  system  r  =  1919487.61+   and  z  =  {M^  ^  1)^  X  1919487.61+,   in  which 

(Jif2  +  1)^  =  [(.4243+)'-;  +  l]""  =  1.086+.  Here  we  have  another  singular  result, 
viz.,  that  the  whole  of  the  infinite  number  of  spires  within  the  value  ?•  =  1  (A  B), 
and  the  spire  generated  by  the  revolution  from  0  =  0  to  0  =  27t,  are  together  only 
a  trifle  longer  than  the  radius  vector  after  this  revolution. 

Ex.  2.  Rectify  the  spiral  of  Archimedes,  ?'  =  -r— . 

Result. — ^For  convenience  put  ~~  =  a,  writing  the  equation  r  =  aB.     Then 

J  dz  =  ~j  (r-  +  a'^)  ^dr,  and  the  process  of  integration  is  identical  with  that  used 

in  rectifying  the  common  parabola  {Ex.  2,  23f)). 

r(a^  +  r'^f    ,    a  ,       f  r  +   (a2  +   r^f 
'  =  2^ +  2^°g 

Ex.  3.  Rectify  the  cardioid,  r  ==  a{l  +  cos  0). 

Solution. — This,  as  the  name  imjjhes,  is  a  heart-shaped  curve.     The  student 
should  first  construct  it.     dr'^  =  a^  sin2  OdQ^,   and  r^  =  a-  -\-  'la-  cos  9  +  a^  cos^  0  ; 

whence  dz  =  {a^  +  2a2cos0  +  «2cos2  9  +  o!2sin2  6)^d9  =  a(2  +  2  cos  9)~d9  = 
2a cos  hOdS,  since  1  +  cos 9  =  2  cos2 19.  (See  Trigonometry  57?  p- )  .•.  z  = 
4:a  sin  ^9  +  C.  Estimating  the  arc  from  9  =  0,  we  have  z  =  0,  whence  (7  =  0; 
and  the  corrected  integral  is  z  =  4a  sin  50.  Making  9  =:  180°  we  have  z  =  4a. 
This  being  ^  the  circumference,  the  entire  length  is  8a. 


-♦-♦-^ 


SUCTION  X. 
Quadrature  of  Plane  Surfaces. 

(a)  BY  KECTANGULAR    CO-OEDINATES. 

244:,  Def. — The  Quadrature  of  a  surface  is  finding  its  area. 
The  term  quadrature  comes  from  the  conception  that  we  find  an 
equivalent  square.  Thus  the  quadrature  of  the  circle  consists  in 
finding  a  square  of  the  same  area. 

24S,  JProp* — The  formula  for  the  quadrature  of  plane  surfaces  is 

dA  =  ydxj  or  xdy. 

Dem. — Recurring  to  Fig.  172,  it  is  proposed  to  find  the  area  of  the  surface  lying 
between  MN  and  AX.  Calling  this  area  ^,  the  trapezoid  PP' D'D  included 
by  two  consecutive  ordinates  is  dA,  a  differential  element  of  the  area  contempora- 


QUADRATURE   OF   PLANE    SURFACES. 


165 


neoua   with    dx,    and   dy,    as   heretofore    considered.       But   area    PP  D  D    ^= 

P  D  -}-  P  D '  1/  -{-  ii  -\-  dy 

X  D  D',  or  dA  =  - — —^^ — -dx  =  ydx  -\-  ^dydx.     Since  the  last 

term  is  a  differential  of  the  second  order  with  reference  to  the  others,  it  must  be 
dropped,  and  there  results  dA  ■==  ydx.  In  like  manner  an  element  of  the  area 
lying  between  a  curve  and  the  axis  of  ordinates  may  be  shown  to  be  dA  =  xdy. 

Q.   E.  D. 


Ex.  1.  Find  the  area  of  the  common  parabola. 


Y 

P 

M    — 

E 

—^ 

A 

\     ° 

D" 

X 

E' 

H 

-.^ 

Fig.  178. 


Solution. — From    y'^  =  1'px,   we    have   dx  =  -ydy. 

1  2/3 

"Whence  dA  =  -y^dy,   and  integrating  A  =  - — j-  C 

Beckoning  the  area  from  the  origin  A,  for  y  =  0,  ^=0, 

whence  (7=0,  and  the  corrected  integral  is  A  =  ^  = 

Sp 

y_i]L—.    P^  —Zg.y;  i.  e.,  the  area  APD  is  lADPE. 

Consequently    PAP'  =  lEPP'E'.      The   latter  rect- 
angle is  sometimes  called  the  circumscribed  rectangle,  and  the  area  of  the  parabola 
is  saiil  to  be  I  of  the  circumscribed  rectangle. 

To  find  the  area  of  any  specified  portion,  as  that  between  x  =  AID  =  a,  and 
X  =  AD"  =  &,  or  of  the  surface   PP"  D"  D,  we  resume  the  indefinite  integral 

3. 

A  =  - — {-  C=  -^ —  -I-  G ;  substituting  a  and  b  successively  for  x,  and  subtract- 
dp  3p 

ing  the  former  result  from  the  latter,  i.  e. ,  integrating  between  the  limits  x  =  a, 

X  =  h,  we  have 


ij 


A  = 


{2pbY  —  {2pay      '  (8)> 


3p 


Sp 


{b'  —J)  =  f(2p)V^— a^). 


Ex.  2.  Eind  the  area  oi  y  =  x 


x\ 


Solution. — Substituting  the  value  of  y  in  the 
general  formula,  it  becomes  dA  =  xdx  —  x^dx 
.  • .  J.  =  5.^2  —  ix-^  -\-  C.  Sketching  the  curve, 
we  observe  that  it  will  be  natural  to  inquire  for 
the  area  AmB,  or  Am'B'.  Thus  reckoning 
the  area  from  the  origin,  x  =  0,  gives  ^=  0,  and 
consequently  C=  0.  The  corrected  integral  is 
A  =  i-r^  —  40;^     Hence  area  A?nB,  or  Am'B'  j,       ^nn 

=  i.     To  find  the  area  of  any  other  portion  as 
BCD,  we  integrate  between  the  limits  a;  =  A B  ==  1,  and  a;  =  A D  =  2. 

gives  A  =  2  — 4:  —  i  =  —  2^.     Hence  area  BCD  =2i. 


This 


Ex.  3.  Find  the  area  oi  y  =  x^  —  b'^x 

Area  betiveen  x  =  0  and  x=b  is  \b^. 


166  PBOPERTIES  OF  PLANE  LOCI. 

Ex.  4.  Find  the  area  of  y  =  x^  +  ax^,  constructing  tlie  curve,  and 
observing  the  natural  Hmits  of  integration. 

BesuU,  Between  x  =  0,  and  x  =  a,  A  =  y^a" ;  and  between  ^  =  0 


and 


00  =  —  a,         dA  =  -^^a*. 


Ex.  5.  Construct  and  find  the  area  of  a^-y"^  =  x^{a'^  —  x'^). 

Area  =  %a^. 

Ex.  6.  Eequired  the  quadrature  of  xy^  =  a^  between  the  hmits 

b  —  c 
y=h,  and  y  =  c.  Area  =  2a^—^. 

Ex.  7.  Required  the  quadrature  of  the  circle. 

1  X. 

Solution.— From  x^-\-y^  =  r%  we  have  y=[r^  —  x'^y,  hence  dA  =  (r^  —  x'^^dx. 

JL 

Applying  formula  <g  of  reduction  and  integrating,  we  find  A  =  ix{r-  —  x-)''^  -}- 
^7-2  gin— il  _|_  C.     Estimating  the  arc  from  the  axis  of  ordinates,  so  that  for  x  =  0, 

1.  X 

^  =  0,  C  is  also  0,  and  the  corrected  integral  is  J.  =  ^x{r"  —  x-)^  -f-  ^r^sin— ^-. 

Making  x  =  r,  we  have  area  of  i  of  the  circle  =  i  r"7t ;  whence  the  entir%  area  . 

of  a  circle  whose  radius  is  r,  is  tct-.     Now  jt  has  been  determined  with  sufficient 

accuracy  in  Ex.  3,  Aet.  230, 

1 
"Without  assuming  the  value  of  7t,  we  may  expand  (r^  —  x-)''^  and  integrate 

approximately.     Thus  dA  =  (r^  —  x^y^dx==rll Ydx,  and  putting  -  =  ajj,  we 

4,  /,      £c,2      .-c  4     X  0    5.-^8     7£c.io     21a-, '2      ^    \, 

have  dA^r-Kl-^,')  d^~=r{l—±~--^^--^^-^--^j^etc.)dx, ; 

whence  A  =  r'^lfdxi  —  ^fxi^dxy  —  ^fx^idx^  —  -i^ef^i^^^i  —  iTEf^x^dx^  — 
iiisfx\  '"tZxi  —  lU-xf^i  ''dxy  —  etc.] 

=  4^^  -  T  -  lo'- 112  -  ri52- 28l6  -133l2  -"*"•]  +  ^' 
in  which  (7  is  0  when  the  arc  is  estimated  from  the  axis  of  y.     Now,  making  x  =  r, 
makes  X]  =  1  ;  whence  the  area  of  i  the  circle  is 

A  =  r-\l  —  -^  —  J-„-  _  -I2-  -  T-,V2-  —  yI-s  —  T:M-F  —  etcV 
The  sum  of  this  series,  thus  extended  is  1  —  .208999  =  .791001.     Hence 

Area  of  circle  =  4.  X  r'^  X  .791001  =  3.164005r^  approximately.  This  number 
3.164005,  if  more  accurately  computed  is  found  to  be  exactly  the  same  as  Tt, 
Ex.  3,  Akt.  239.     A  nearer  approximation  can  readily  be  made  by  extending  the 


series. 


24:6,  CoR.  1. — The  areas  of  circles  are  to  each  other  as  the  squares  of 
their  radii. 

Dem.— Let  r,  and  r^  be  the  radii  of  two  circles.     Their  areas  are  TCr^  and  jtri^. 
But  Ttr^  :  TCri  2  : :  r^  :  r ,  2. 

24:7 •  Cob.  2. — The  area  of  a  circle  whose  radius  isl  is  tt. 


QUADRATURE   OF   PLANE   SURFACES.  167 

1^4:8 •  Cor.  3. —  The  area  of  a  circle  is  equal  to  its  circumference  into 
J  its  radius. 
Dem.     r^7t  =  Irit  X  kr,  and  2r;r  is  the  circumference  {239 ^  Ex.  3). 

24:0,  I*TOh. — To  find  the  area  of  any  segment  of  a  circle. 

Solution. — Let  the  distances  from  the  centre  to  the  bases  of  the  segment  be  6 
and  a,  so  that  a  —  6  =  the  height  of  the  segment.  We  have  but  to  resume  the 
indefinite  integral  and  take  its  value  between  these  limits .    Thus 


/. 


dA  =  \a{r'^  —  a^-f  -}-  2-r2  sin-' ^h{r^  —  h'^)    —  ^r'-  sin-i-. 

6 


If  the  segment  is  reckoned  from  the  diameter,  5=0,  and 


X-  a 


I      dA  =  ^a{r-  —  «>)^  -{-  -i-r-sin-'-.     The  significance  of  this  is  readily  seen  by 
«^=  0 
inspecting  Fig.  174.     AD  =  a,  and  j^a{r^  —  a^)    =  area  of  APD.     Arc  BP  = 

ci  a 

r  sin— 1—  ;  whence  -ir^  sin— i-  =  area  sector  A  B  P. 
r  r 

If  the  segment  has  but  one  base,  a,=  r,  and  we  have 
/     dA  =  |7rr2  —  ^hir-  —  \fif  —  -^r^sin-'-.     From  Fig.  174,  we  see  that  krcr^  =  | 


'• 


the  area  of  the  circle,  |-r2sin— i-  =  area  BPA,  and  ib(r^ —  &2)2  __  q^q^^  APD. 

r 
In  each  case  we  have  |-  the  segment. 

Ex.  8.   Find  the  area  of  the  ellipse. 

Sug's.— The    area=-/    {A^  —  x'^)''^ dx.    But    /    (^2  —  cc2)2'(;^x  is  i  the  area  of 

a  circle  whose  radius  is  A,  which  is  vtA^.     .  * .  Area  of  ellipse  =  itAB. 

2S0»  CoK. — The  area  of  an  ellipse  is  to  the  area  of  the  circumscribed 
circle  as  the  conjugate  axis  is  to  the  transverse  axis.  The  area  of  an  ellipse 
i",  to  the  area  of  the  inscribed  circle  as  the  transverse  axis  is  to  the  conjugate 
axis.  The  area  of  an  ellipse  is  a  mean  proportional  between  the  inscribed 
and  circumscribed  circles. 

Ex.  9.  Find  the  area  of  the  cycloid. 

Sitg's. — We  have  dx  =  ^-^- r  ;    whence  dA  =  ~  ^  — 


L 


(2ry  —  yrf  (2ry  —  y^)^ 

S.  _JL 

y^{2r  —  y)  ''^dy.     Integrating  by  applying  formula  ^  twice,  we  obtain 


/*2r  s. 


2r  3.  _1 

(2r  —  y)  *(i?/  =  fr2Ters— '2 


=  f r^TT,  since  vers— '2  =  tc.  .  • .  The 
entire  area  is  STtr"^,  or  three  times  the 
generating  circle. 

A  somewhat  indirect  but  simple 
method  of  quadrature  is  as  follows  : 

1st.  Find  the  area  of  APCB,  the 


Y 

B 

DD' 

c 

r 

7" 

^ 

\ 

^ 

N 

A 

/ 

Fig. 

180. 

E  X 

168  PBOPEBTIES  OF  PLANE  LOCI. 

element  of  whicli  DPP'D'  =  dx{2r  —  y)  —  {2r  —  y) ^-^ — -  —  {^ry  —-fydy. 

Now  considering   the   circle   A'pC,   we   observe   that   an   element  as  pdd'p'   = 

dy{2ry  —  ?/-)^.  .•.  APOB  =^^  area  of  the  generatrix,  =^7tr-.  But  ABC  A' = 
2r  X  A  A'  z=  2r  X  7tr  =^  'litr-.  .-.  ACA'  =  fTrr-,  or  the  entire  area  of  the 
cycloid  :=  37trK  Observe  that  both  these  integrals  are  to  be  taken  between  the 
same  limits,  viz.,  y  =  0,  and  y  =^  2r. 

Ex.  10.  Find  the  area  of  the  curve  a^y~  =  a-b^a:-^  —  b'^x'^. 

Area  =  ^ab. 


(6)    QUADKATUEE   OF   POLAil   CUKVES. 

2S1,  JPvoj), — An  elemerit  of  the  area  of  a  polar  curve  i^  -i-r^d^?  and 
the  formula  for  the  quadrature  is     dA=^r^d.d. 

Dem.— In    Fig.    176    PAP'  =  dA.      But    area    PAP'  =    AP'  X  iPD  = 

(?•  -\-  dr)jrdS  =  jrr-dQ,  omitting  irdrdB,  and  also  remembering  that   P  D   =  rd6. 
.  • .  dA  =  ^r^dQ.     Q.  e.  d. 

Ex.  1.  Eind  the  area  of  the  spiral  of  Archimedes. 

Solution.  ■ — The   equation  is  r  =  ^r— 0  ;    whence  dr  = 

27r 

— -cZ9,  and  d6  =  27tdr.    Substituting  in  the  formula,  we  have 

dA  =  Ttr^dr,  and  A  =  iTtr^  -f-  O  K  we  estimate  the  area 
from  the  pole,  we  have  4  =  0,  when  r  =  0,  and  conse- 
quently C  =  0.  The  corrected  integral  is,  therefore,  A  = 
iTtr^.  This  is  the  general  expression  for  the  area  passed  over  by  the  radius  vector  in 
its  revolution  from  its  starting  at  0  to  any  value,  as  r  At  the  end  of  one  revolution 
J'  =  AP  =  1,  and  A  =  area  of  the  first  spire  =  AmP  =  i;r=  i  the  area  of  a 
circle  described  with  A  P  as  a  radius.  At  the  end  of  the  second  revolution  r  =;  2, 
and  the  area  traced  by  the  radius  vector,  or  Aj  =  ^tc.  This  evidently  includes 
twice  the  first  spire  -\-  the  second.  The  area  of  the  second  spire  is,  therefore 
f  tT  —  ^Tt  =  27t.     The  area  of  the  first  two  spires  is  ^tt. 

Ex.  2.  Show  that  the  area  of  the  Napierian  logarithnTic  spiral  is  ^- 
the  square  described  on  the  radius  vector. 

Ex.  3.  Find  the  area  of  the  Lemniscate  of  Bernouilli.   (r2=  a-cos  Id. ) 

SoLTJTioN. — From   the   formula  we  have   dA  =  ^a^cosWdB.      "Whence   A    = 

ia2  /     cos  20  dS  =  ia2sin26  =  la'^     .  • .  The  entire  area  is  a%  i.  e.,  the  square  on 

•the  semi-axis. 

Ex.  4.  Trace  the  curve  r  =  a(cos  26  -f  sin  2^),  and  find  its  area. 


QUADRATURE  OF  SURFACES  OF  REVOLUTION.         169 

SUG*S.     r'?  =  a2(cos2  29  +  sin2  29  +  2  sin  26  cos  20)  =  a\l  +  sin  40)      - .  A  == 
ri(j2(i  _j_  sin40)d0  =  ^a^{fdd  4-J"sin40  cZ0}.     The  entire  area,  which  is  com- 
prised between  9  =  0  and  0  =  27t,  is  Tta'^. 


^♦» 


SUCTIOJSr  XL 

Quadrature  of  Surfaces  of  Eevolution, 

2,S2,  Def. — A  Surface  of  Mevolution  is  a  surface  gener- 
ated by  a  line  (right  or  curved)  revolving  around  a  fixed  right  line  as 
an  axis,  so  that  sections  of  the  volume  generated  made  by  a  plane 
perpendicular  to  the  axis  are  circles. 

Ill's. — A  right  line  parallel  to  and  revolving  around  another  right  line,  in  the 
manner  described  in  the  definition,  generates  a  cylindrical  surface  with  a  circular 
base.  A  semi-elhpse  revolving  around  its  transverse  axis  generates  a  prolate 
spheroid,  around  its  conjugate  axis  an  ohlate  spheroid.  These  two  are  varieties  of 
ellipsoids.  A  paraboloid  is  generated  by  the  revolution  of  a  parabola  around  its 
axis.  The  number  of  these  surfaces  is,  of  course,  infinite  ;  and  the  specific  char- 
acter of  each  depends  upon  the  nature  of  tne  generating  curve. 

2S3,  I^TOp, — The  differential  element  of  a  surface  of  revolution  is 

dS=  27ry\/dy'  +  dx\ 

Dem. — Let  M  N  be  the  generatrix,  AX  the  axis  of  rev- 
olution and  P  P'  two  consecutive  points  on  the  generatrix. 
As  M  N  revolves  about  AX,  any  point,  as  P,  whose  co- 
ordinates are  x  and  y  describes  a  circle  whose  circumference 
is  27ty,  The  circumference  described  by  the  consecutive 
point  P'  will  be  27t{'i/  -f-  dy).  PP'  describes  the  frustum 
of  a  cone  whose  surface  is  half  the  sum  of  the  circumfer-  jij^^^  3^32. 

ences   traced   by    P   and    P'    multiplied   by    PP',  or  = 

y  -\-  ^^  +  ^  y  ,/cly2^clx^^  Whence  omitting  2itdy,  it  being  an  infinitesimal 
of  a  higher  degree  than  the  other  terms,  we  have  dS=  27Cy\/dy'^-\-dx^.     q.  e.  d. 

ScH. — To  apply  this  formula,  let  y  =  <p{x)  be  the  equation  of  the  genera- 
trix, from  which  find  dx  or  dy,  and,  having  substituted  in  the  formula, 
integrate  between  the  proper  limits. 

Ex.  1.  Find  the  area  of  the  surface  of  a  sphere. 


Solution.  —From  y'^  -{-  x^=z  r^  we  have  dy^  =  — —.     .* .  dS==  ^Tty    j  dx^  -\- " — j- 

y  ^^  if 

=  27trdx  ;  and  6'  =    /    27trdx  =  2;rr2.     This  being  the  surface  of  a  hemisphere 
the  entire  surface  of  the  sphere  is  47rr2. 


c 


170  PEOPEETIES   OF   PLAls'E   LOCI. 

CoE.  1. — Since  7tr^=  area  of  a  great  circle,  the  Y 

surface  of  a  sphere  =  4  great  circles.  Again,  since 
4,7tvi  =  27rr  X  2r,  the  surface  of  a  sphere  =  the 
circumference  of  a  great  circle  X  the  diameter. 

A  DE      B  X 

2d4:.  Cor.  2. — The  portion  of  the  surface  gen-  Fig.  183. 

erated  by  any  arc,  as  P  S  is  a  zone.  To  find  the  surface  of  a  zone,  let 
A  D  =  a  and  A  E  =  b  and  integrate  dS  =  27rrdx  between  these  limits  ; 

thus,  S  =    /  27tvdz  =  27rr(b  —  a)  ;    i.  e.,  the  surface  cf  a  zone  =  its 

altitude  multiplied  by  the  circumference  of  the  great  circle  of  fhe  sphere. 
Hence,  also,  the  surfaces  of  zones  on  the  same  sphere  are  to  each  other 
as  their  altitudes. 

ScH. — If  a  cylinder  be  circumscribed  about  a  sphere,  its  convex  surface 
is  the  circumference  of  its  base  (27Cr)  X  by  its  altitude  (2r)  ;  i.  e.,  4:7tr% 
the  same  as  the  surface  of  the  sphere.  Add  to  this  27tr%  the  upper  and 
lower  bases,  and  the  entire  surface  of  the  cylinder  is  found  to  be  67tr%  The 
surface  of  the  cylinder  is,  therefore,  to  the  surface  of  the  sphere  as  3  to  2. 
This  fact,  together  with  the  same  relation  between  the  volumes,  was  dis- 
covered by  Archimedes,  and  the  discovery  was  so  much  thought  of  by  him 
that  he  expressed  a  wish  that  the  device  on  his  tombstone  might  be  a  sphere 
inscribed  in  a  cyHnder.  The  great  geometer  was  murdered  by  the  soldiers 
of  Marcellus,  B.  C.  212,  though  contrary  to  the  orders  of  that  general. 
Marcellus  executed  the  device  of  the  sphere  and  cyhnder  upon  the  tomb, 
and  buried  the  philosopher  with  honors.  140  years  after,  when  Cicero  was 
questor  in  Sicily,  the  place  of  the  grave  had  been  forgotten  ;  but  he,  chanc- 
ing to  remember  the  device  upon  the  stone,  sought  out  and  restored  the 
tomb,  which  had  become  overrun  with  weeds  and  thorns. 

Ex.  2.  To  find  the  area  of  the  surface  of  the  paraboloid. 

SoiiXTTioN. — From  y^  =  2px,  we  have  dx^  =  —~.     .'.  dS=  —y(p^  -f  y^)^dy, 

and  S  =    r^(p^  +  y-)  dy  =  ^(P^  +  7/^)^  +  C.     Eeckoning  S  from  y  =  0, 


Y^y  -^^  '  -^  -  Zp' 


27r  3. 

C=  —  f7rp2  and  the  corrected  integral  is  S  =  -   [(p2  _|_  ^/i)^  — ps]^  which  may  be 

satisfied  for  any  value  of  y. 

Ex.  3.  To  find  the  area  of  the  surface  generated  by  the  revolution  of 

2  2  2 

the  hypocycloid  x^  -\-  y^  =i  6^  about  the  axis  of  x. 

1    A    2  12 

S  =  27vc^  I    y'^  dy  =%7tc^ ;   whence  2S  ==  -^,  the  area  of  fhe 

fixed  circle. 

Ex.  4.  Find  the  area  of  the  surface  generated  by  the  cycloid  revolved 
about  its  base. 

Sug's.     dS  =  27rr(2r)  y(2r  —  y)^dy,  which    integrated  by  formula  il  is  jj?  « 


OUBATUKE  OF  VOLUMES  OF  REVOLUTION.  171 

—  f 7r\/2r2/(2r  —  y)-  —  i/;r?V2r(2r  —  yy  +  G.  This  taken  between  y  =  2r,  and 
y  =  0  gives  for  half  the  surface  ^^7tr%  and  for  the  whole  surface  V"  times  the  gen- 
erating circle. 

Ex.  5.  Find  the  area  of  the  surface  of  the  prolate  spheroid. 

Solution.— From    a^y~   +   h^x-  =  a^b'^    we    get    dy^  = dx^.       .-.    dS  = 

2Tty(p^~^^ydx  =  ~[a2(a262  —  y^^)  -f  h^x^fdx  =  —{a^  —  a2e^x-^)^dx  = 

lTt-(Gfi — €^x^)^dx.    "Whence  half  the  surface  of  the  ellipsoid  =  2;r-   /  {a^  —  e^ic^)  ^dx. 

i-  ^  L 

To  integrate  {a^  —  e^x~Ydx  apply  formula  ^^  which  gives  J  (a'^  —  e^x^Y'^dx  = 

-^^ ^-^ =  i^{a^  -  e^x^f  +  W  / j:  = 

J  {a"  —  e'^x'^f 

•1-  ni  px  7)1"  i  /7-  />'>•  ~l 

ia;(a2  _  ezx^y  +  i_  sin->  -  +  C.  .  • .  ^S  ==  tt-  a;(«2  —  esa;^)^  +  -  sin-'  -  +  (7  . 
Eeckoning  ;S^  =  0  for  ic  =  0,  C  =  0  ;  and  then  putting  a;  =  a  we  have  for  half  the 

X  1  Ttdh 

surface  7tdb[(l  —  e-)-  -\ — sin— 'e],  or  nh^  -\ sin— le. 

6  6 

Ex.  6.  Apply  the  above  formula  to  find  the  area  of  the  surface  of 
the  prolate  spheroid  whose  generatrix  is  25?/^  +  IGor^  =  400. 

Area,  235.41  nearly. 


^  ♦ » 


SECTION  XIL 
,  Oubature  of  Volumes  of  Eevolution. 

2SS,  J^TOp, — The  differential  element  of  volume  of  a  solid  of  revo- 
lution is  dV  =  ;ry2dx. 

Dem. — Using  Fig.  182,  with  the  same  notation,  the  volume  generated  by  the  revolu- 
tion of  DPP  D'  about  D  D',  is  a  frustum  of  a  cone,  and  is  therefore  equal  to  three 
cones  having  for  their  common  altitude  the  altitude  of  the  frustum  {dx),  and  for  bases 
the  upper  base  (7ty^),  the  lower  base  l'!t(y-\-dyY'\,  and  a  mean  proportional  between 

XI     j^      ,  r     /      .    1    -,      Tr  -rrr      It' y^ -\-y^ -\-1ydy -\- dy^  A-v'^ -\-ydy)dx 

the  two  bases  \7ty{y  +  dy)\     Hence  dV  —      -^^^^^^^-^    lyry-/ —  _ 

o 

(omitting  terms  having  infinitesimals  of  a  higher  order)  ity'^dx,     Q.  E.  d. 

-  Ex.  1.  To  find  the  volume  of  a  sphere. 
Solution,     y^  z=  r^  —  x\     .- .  dV  =  7t{r'^  —  x^)dx  =  Ttr'^dx  —  itx'^d^,  V  = 
I    {Ttr^dx  —  Ttx'^dx)  =  f  ^7*3 ;  or,  letting  D  =  the  diameter,  and  doubling  for  the 
entire  volume  F=  knD'^ 


172  PROPERTIES  OF  PLANE  LOCI. 

2d 6,  Cor.  1. — Since  ^tti^  =  ^r  x  4i7rY%  the  volume  of  a  sphere  ==  the 
surface  X  ^  the  radius. 

2S7»  CoK.  2. — The  volumes  of  spheres  are  to  each  other  as  the  cubes 
of  their  radii  or  their  diameters. 

2SS.  CoE.  3. — The  portion  of  a  sphere  included  between  two  parallel 
planes  is  called  a  segment.  If  a  and  b  represent  the  distances  of  these 
planes  from  the  centre^  the  volume  of  the  segment  is 

Y  =  7r[v^{h  —  a)  —  i(b3  —  a^)]. 

239,  Cor.  4.  f  tti-^  =  §of  Ttv^  x2t  =  ^  of  the  circumscribed  cylinder. 
See  Scholium  under  Art.  2S4:, 

Ex.  2.  Find  the  volume  of  the  Prolate  Spheroid.  Also  of  the  ob- 
late Spheroid.  Show  that  each  is  |-  of  the  circumscribed  cylinder. 
Deduce  from  each  the  volume  of  the  sphere. 

Ex.  3.  Find  the  volume  of  the  paraboloid. 

Volume  =  one  half  the  circumscribed  cylinder. 

Ex.  4.  Find  the  volume  of  the  cissoid  revolved  about  its  asymptote. 

Solution. —Let  the  curve  AM  revolve  about  BT, 
then  will  PDP'  D'  be  an  elemental  section  of  the  vol- 
ume if  DD'  =  cZ?/.  PD  =  2a  —  x,  and  the  circle 
traced  by  P  D  in  its  revolution  is  ;r(2a  —  xy.  There- 
fore the  element  of  volume  or  d  V=  tt (2a — x)  ^dy.     From 

the  equation  of  the  curve  y^  =  - — - — -,  we  have  dy  = 

X 

3a.r2  —  x^ ,  (3a  —  x){1ax  —  x'^f  ,        „,, 

— -dx  = dx.     Whence  sub-  Fig.  184 

i-  i  X  a  i 

stituting,  dV=  7t{3a  —x){2ax  —  x'^^dx,  =  Za7tx^(2a  —  xYdx  —  7tx^ {2a  —  x)'^dx. 

These  terms  may  be  integrated  by  the  use  of  formulas  ^  and  <^,  and  we  find 
^a7tx^{2a  —  x)'^dx—  j    7tx\2a  —  xf dx  =  jt^a\ 

»/o 


<♦♦»■ 


m/. 

T 

Y 

/ 

/^/ 

K 

D' 

\ 

/  p/ 

\ 

D 

IJ'           \ 

A 

E 

I      X 

SECTION  XIIL 

Equations  of  Curves  deduced  by  tlie  aid  of  the  Calculus. 

2(y0,  Def. — Tlie  Tractrix  or  Equitangential  Curve  is  generated 
by  the  motion  of  a  weight  drawn  upon  a  plane  by  a  cord  of  constant 
length,  the  extremity  of  the  cord  moving  along  a  straight  line  not  in 
the  direction  of  the  cord  itself.     The  portion  of  the  tangent  inter- 


EQUATIONS  OF  CUEVES  DEDUCED  BY  THE  AID  OF  THE  CALCULUS.   173 

cepted  between  the  curve  and  the  fixed  hne,  is  a  constant  quantity 
(the  length  of  the  string),  and  hence  the  second  appellation. 

ItiTjUS. — Let  a  weight  be  placed  at  B,  Fvj,  185,  with 
a  string  of  the  length  A  B  attached.  As  the  extrem- 
ity of  the  string,  A,  is  carried  along  the  line  AX,  B 
will  trace  BM,  the  iractrix.  (This  conception  sup- 
poses friction  to  exist  but  not  momentum.)  "When 
the  extremity  of  the  string  is  at  any  point  in  the  line,  '~fi\  d~o*  t^ 
as  "T,  the  weight  will  be  at  P,  and  it  is  evident  that  Fig.  185. 

PT",  the  tangent,  will  be  constant. . 

23 1.  I^TOh, — To  find  the  equation  of  the  Tractrix. 

Solution.— Let  P  and  P'  be  consecutive  points  on  the  curve,  PD  =  y, 
PE  =  —  dy,  (minus  since  y  decreases  as  a;  increases),  AD  =  a;  and  D  D'  =  dx. 
Let  AB  =  PT  =  a.     Then  DT  =  v/«2  —  yi,  and  PE  :  EP'  ::  PD  :  DT, 

or  by  the  notation  —  dy  :  dx  :  :  y  :  (a'^  —  2/'")  •     •  *  •    i^  ~\ =  0- 

ClX  s- 

(a2_  2/2)2 
Or  this  differential  equation  may  be  obtained  from  the  general  differential  value 

of  the  tangent ;  thus  (l-\-~  yy  =  a,  whence  -^  =  zh .     The  -J-  sign 

indicates  that  the  curve  is  generated  by  motion  to  the  left,  as  then  tan— i  --  is  + 

dx  ' 

and  the  —  sign  is  to  be  taken  when  the  curve  is  generated  as  in  the  illustration  above. 
In  order  to  have  the  equation  in  finite  terms  it  remains  to  integrate  this  differential 
between  proper  hmits.  The  inferior  limit  of  x  is  evidently  0,  to  which  y  =  a  cor- 
responds.   The  superior  limit  must  be  left  general,  as  the  curve  extends  to  infinity. 

J. 

(^2   ^2)2  /•■e 

Putting    the    equation   in    the    form  dx  =  —  -^ — dy  we  have      /    dx  = 

y  Jo 


'-—1-Ldy  =  -        ( 1-  M=alog. 

^  Ja    ^i;ra2_w2^2        c«2_„2>>2^ 


y{a^  —  y^Y       {a^  —  y^f^  ^ 

(«2  —  y2y.     .'.  The  equation  is  a;  =  « log •— (a^  —  y^y. 


262,  JPvoh, — To  find  the  equation  of  a  curve  whose  subnormal  is 
constant. 

Solution. — The  general  differential  value  of  the  subnormal  is  t/--.     .*.  Letting 

dx 

p  be  the  constant  value  of  the  subnormal,  we  have  y-j-  =p,  or  ydy  =  p  dx.  In- 
tegrating, ?/■-  =  2px  -f-  G'  This  is  the  equation  of  the  parabola,  as  it  should  be, 
since  the  constant  subnormal  characterizes  that  curve.  G  is  not  determined  by 
the  problem.  If  the  condition  "which  passes  through  the  origin,"  were  added 
to  the  problem,  C  would  be  0. 


174  PROPERTIES   OF   PLANE   LOCI. 

203,  JProb, — To  find  the  equation  of  the  curve  whose  normal  is  of 

constant  length. 

ax'- J 

ting  this  equal  to  the  constant  r,  we  have  y\\-X-  -^  )    =  r,  or  —  = 1,  or 

\         dxV  dx^       j/"2 

dx  = ^ j^  =  y{r^  —  2/^)  *(^2/-     Placing  the  origin  on  the  curve,  so  that  the 

inferior   hmit  of  the  integral   shall   he  x  =  0,  y  =  0  and  we  have    I   dx  = 

2/(r2  —  y^)  ^dy,  or  a;  =  r  —  {r^  —  y'^)'^   whence  ?/2  :=:,  ^rx  —  x^.      This  is  the 

well  known  equation  of  the  circle,  as  it  should  be,  the  normal  of  the  circle  being 
its  radius,  and  hence  constant  : 

204.  I^voh, — To  determine  the  equation  of  the  curve  whose  subtan- 
gent  is  constant  (m). 

m  log  y  =  X  +  C.      The  logarithmic  curve. 

2SS,  I^TOb, — To   determine  the  equation  of  the  curve  ivhose  sub- 
normal varies  as  the  square  of  the  abscissa. 

y2  =  2.X3  _|_  c.     The  semicubical  parabola. 


206.  ^TOh. — To  determine  the  equation  of  a  curve  such  that  the 
area  shall  equal  twice  the  product  of  its  co-ordinates. 

SoLimoN.     jydx  =  2xy,  or  ydx  =  2xdy  -}-  2ydx  or  —  =  —  i — .     Integrating, 

y  X 

logy  =  —  \\ogx -{-C=  —  log.'j;^  -f- ^-     •*•    C=logxy.     As  log a;^2<' is  constant 

x'^y  must  be  constant ;  therefore  we  may  put  x^y  =  m,  or  xy'^  =  m'^  is  the  equa- 
tion sought. 


207.  J^rob. — Find  the  equation  of  the  curve  whose  arc  varies  as  the 
square  root  if  the  third  power  of  the  abscissa. 

Solution.    J  (dx^  +  dy^)-  ^^  ^x^,  using  ?7i  for  any  constant  factor.     Eemovmg 
the  sign  of  integration  by  differentiation  and  squaring  both  sides  we  have  dx'^  -f-  dy'^ 

1  8  1 

=  fm2xda;2,  or  dy  =  (f m^x  —  1 ) '^ dx ,    Whence  integrating,  y  =  <c=— ;( I w^a;  —  If  -{- C, 

the  semi-cubical  parabola. 


OF  TANGENTS  AND  NOEMALS. 


175 


SUCTIOI^  XIV. 
Of  Tangents  and  Normals. 

[WITHOUT  THE  AID  OF  THE  CALCULUS.] 

[Note. — Students  taking  a  shorter  course,  witliout  the  Calculus,  will  omit  the  thirteen  preceding 
sections  of  this  chapter,  and  conclude  the  course  with  this  and  the  following  section.  Such  as 
take  the  fuller  course  will  not  need  to  read  this  section,  but  should  read  Sec.  XV.] 

2GS     T^TOh, — To  produce  the  equation  of  a  tangent  to  a  plane  curve. 

Solution. — Let  M  N  be  any  plane  curve  whose 
equation  is  y  =f<^x)*,  ox  fyx,  y)  =  0.  Let  P'be  a 
point  in  the  curve  whose  co-ordinates  are  (x',  y'), 
and  P"  a  point  whose  co-ordinates  are  (x",  y"). 
Then    the    equation   of   the   secant  line    RT"   is 

y'  —  y" 


y  —  y 


—  {x  —  x)  {31).     Now  as  P' and 


X   —  X 

P'  are  points  in  the  curve  their  co-ordinates  will 
satisfy  the  equation  of  the  curve,  and  we  have 

(1)  y  =f{x')  and  y"—f(x");    or 

(2)  fix;  y')=  0  and  /,x",  y')  =  0. 


Fig.  186. 


y 


y 


If  now  we  can  find  the  value  of  —, -—  from  the  two  equations  (1)  or  (2),  in 

such  a  form  that  it  will  take  a  determinate,  finite  form  when  y'  =  y",  and  x'  =  x", 
that  is  when  P'  and  P"  coincide,  this  value  substituted  in  the  equation  of  the 
secant  will  transform  it  into  the  equation  of  a  tangent,  since  it  is  evident  that  as 
P'  and  P"  approach  each  other  the  line  RT  passing  through  them  approaches  a 
tangent,  and  becomes  a  tangent  when  the  points  coincide,     q.  e.  d. 

Ex.  1.  Produce  the  equation  of  a  tangent  to  tlie  parabola. 

Solution. — Letting  {x,  y')  and  {x"  y")  be  two  points  on  the  curve,  the  equa- 

v'  —  y" 

tion  of  a  line  passing  through  them  is  y  —  2/'  =  '—- 'tjt'k''^  —  ^')j  which  is  the 


X  —  X 

equation  of  a  secant,  since  the  two  points  are  in  the  curve.  Moreover  as  {x' ,  y') 
and.(x",  y")  are  in  the  curve  they  satisfy  the  equation  of  the  curve  ?/' =  2px, 
giving  (1)  y"^  =  2px',  and  (2)  y"^  =  2px".  Subtracting  (2)  from  (1)  y'~  —  y"-2  = 
2p{x'  —  X"),  or  dividing  by  x  —  x",  and  y'  -f-  y",  we  have 

—-  =.  — — ,  =  —,  when  the  point  (x",  ?/")  is  made  to  coincide  with 

x'—x"     y'-\-y"      y  r       v    .^  / 

{x',  y).     Substituting  this  value  in  the  equation  of  the  secant,  we  have  for  the 


P  ,  / 

tangent  y  -  y'  =.—,{x  —  x),  ox  yy 


y'-2  =px  — px,  or  yy'  =  y''^  -f-  px  — px'. 


*  This  is  read  "  y  equals  a  function  of  x,"  and  is  simply  a  general  form  comprehending  every 
equation  containing  x  and  y,  supposed  solved  in  respect  to  y.  The  second  form,  read  "  function 
of  X  and  y  =  O.''  indicates  the  same  thing  except  that  the  equation  is  not  supposed  to  be  solved. 


176  PBOPERTIES  OF  PLANE  LOCI. 

But  as  {x',  2/')  is  a  point  in  the  curve  y'^  =  2px'.     Substituting  this,  and  uniting 
terms  we  have  yy'  =  p{x  -\-  x). 

Ex.  2.  Produce  the  equation  of  a  tangent  to  the  parabola  whose 
parameter  is  9,  at  ^  =  4.  Construct  the  tangent  from  its  equation, 
and  then  construct  the  parabola,  thus  observing  that  the  right  line  is 
tangent  to  the  curve  at  the  given  point. 

Solution. — [We  might  proceed  as  in  the  general  method,  but  it  will  be  better  to 
use  the  equation  of  the  tangent  to  the  parabola,  both  because  it  will  require  less 
work,  and  because  it  is  important  that  this  form  should  be  made  familiar.] 

The  equation  of  this  parabola  is  y^  =  ^x.  For  a;  =  4,  2/  =  zb  6.  Taking  the  point 
(4,  6)  we  have  x'  =4,  and  y'  =6.  Hence  yy'  =^p{x-\-x')  becomes  &y=^4^{x  -}-  4), 
ox  y  =  Ix  A-  3,  as  the  equation  of  a  straight  line  which  is  tangent  to  the  parabola 
2/2  =  9ic  at  £C  =  4,  2/  =  6. 

Construction. — 1st.  Constructing  the  line  y  =  ix  -\-  3, 
we  find  that  it  cuts  the  axis  of  y  at  (0,  3) ,  that  is  at  C, 
and  the  axis  of  x  at  ( —  4,  0),  that  is  at  T*.  "Whence  we 
draw  RX.  2nd.  Constructing  the  parabola  y^  =  9x, 
we  find  that  the  hne  RT  touches  it  at  P,  whose  co- 
ordinates are  (4,  6). 

As  a;  =  4  gives  ?/==  d=  6,  we  have  another  point  (4,  — 6) 
which  is  embraced  by  the  example.  The  equation  of  a 
tangent  at  this  point  is  y  =  —  |x  —  3.  [Let  the  student 
produce  and  illustrate  it.  ]  Fig.  187. 

Ex.  3.  Produce  the  equation  of  a  tangent  to  the  parabola  whose 
parameter  is  10,  at  x  ==  10,  and  construct  and  illustrate  as  above. 


Ex.  4.  Produce  the  equation  of  the  tangent  to  the  ellipse  referred 
to  its  axes. 

v'  —  v" 

SuGs. — Equation  of  secant,  y  —  y' ^—, -—{^  —  x').     The  equation  of  the 

ellipse  A^y'^  -f-  B^x^  =  A^B^-,  satisfied  for  the  points  {x ,  y')  and  {x",  y"),  gives 

(1)  A^y'^    +  B^x'^    =  A^B^  and 

(2)  A2y"-2  -f  B^x'"^  ±=  A^B\ 

Subtracting,         A\y''2  —  ^Z"')  -{-  B\x'-^  —  x"-)  ■=:0',  whence 

y'—y"  B^/x  4-x"\  B^x'      ,        ,,,,..-,  .     .-,     X 

-; r,  = rri   tH — 77  I  = 77-7j  wheu  (x  ,  11  )  becomcs  comcident 

X  —X  A\y^y  /  A^y  ^       ^  ' 

with  {x' ,  y').     Hence  the  equation  of  the  tangent  is 

B^x' 
y  —  y  =  —  "42^"^^  —  ^')  '  '^^j  clearing  of  fractions  and  transposing, 

A^yy'  4-  B'^xx'  =  A^y'^  -f  B^x'^.  But  since  {x',  y')  is  in  the  curve  A^'^  -\-  B^x'^  — 
A^B^,  which  value  substituted  gives  for  the  equation  of  th«»  tangent  to  the  eUipse 
A^yy'  -f-  B'^xx:  =  A^B-2. 


TANGENTB.  177 

Ex.  5.  "What  is  the  equation  of  a  tangent  to  the  ellipse  whose  axes 
are  6  and  4,  at  a;  =  2  ?     Construct  as  above. 

Ans.,   Calhng   the   point  of    tangency    (2,    1.49)   the    equation  is 

2/  =  —  .5966^  +  2.66.     Calling  the  point  of  tangency  (2,  — 1.49) 

the  equation  is  y  =  .596607  —  2.66. 

Ex.  6.  Produce  the  equation  of  a  tangent  to  dy^  -f  x'^  =:  5,  at  j;  =  1. 
Construct  and  illustrate  as  before. 

SuG. — See  Ex.  2,  page  97,  second  solution. 

Ex.  7.  Show  that  the  form  of  the  equation  of  a  tangent  to  a  circle 
at  a  given  point  {x',  y')  in  the  circumference  is  yy'  +  ^^'  =  -S^- 

Ex.  8.  What  is  the  equation  of  a  tangent  to  a  circle  whose  radius 
is  5  at  J7  =  3  ?  Equation,  3/  =  q=  f  ^r  db  6^^. 

Ex.  9.  What  is  the  equation  of  a  tangent  to  a  circle  whose  radius 
is  10  at  ^  =  —  3?     At  2/ =  —  4?     At2/  =  1? 

One  equation  is  y  =  da  ^v^lx  —  25. 


Ex.  10.  Produce  the  equation  of  the  tangent  to  the  hyperbola. 

Equation,  A^yy'  —  B'-xx'  =  —  A^B\ 

Ex.  11.  Produce  the  equation  of  a  tangent  to  the  hyperbola  whose 
axes  are  4  and  2,  at  ^  =  4.     Construct  and  illustrate. 

The  points  of  tangency  are  (4,  v/3),  and  (4,  —  \/3). 

Equation  of  tangent  at  (4,  v  3),  y  =  ^vZx  —  -g-v  3. 

(4,  —  \/3),  2/ =  —  iv/3a;  +  ^v/3. 

Ex.  12.  Produce  the  equation  of  a  tangent  to  ^y^  —  2.^2  =  10, 
at  a?  =  4.     Is  there  more  than  one  tangent  ?     Construct  as  above. 

Equation,  ?/  =  ±  .7127.2;  db  .8909. 


x^ 
Ex.  13.  Produce  the  equation  of  a  tangent  to  y^  = ,  at  ^  =  2. 


'3  .7:";^ 


cc  •'  ,      ,,  !K 


Sug's. — Reasoning  as  on  Ex.  1,    we  have  y''^  = -^,  and  y"'^  z= 

Subtracting,  y'^  —  y"-^ 


4  —  X  4  —  x' 

x''^(4:  —  a;")  — .r"^(4  —  x')      4x'3  — x'^'x"  — 4ic"^  -\-x"^x' 


(4  —  x'){4:—x")  (4  —  cc')(4  —  x") 

Mx'3  —  .r"3)  —  x'x"{x'^  —  a;"2) 


(4  —  x'){4.  —  x") 


Whence  dividing  by  y'  -\- y" ,  and  x  —  x" ,  we 


V  — w         4('cc2-f-xfc    +  x  2)  —  XX  {x  -\-x   )       Qx'^ — .r  ^ 

have = ■ ; r,— ! =  -7—, T-  when  x    =x  . 

X  — X  (2/'  +  2/   )(^  —  ^'K^  —  ^   )  2/ (^  —  ^y^ 

v'  —  v" 

and  y"  =y' .    Substituting  this  value  of  —, -r,  m  the  general  equation  for  the 


178  PROPERTIES  OF  PLANE  LOCI, 

secant,  we  have  for  the  general  equation  of  a  tangent  to  this  locus  y —  y'  = 
6a;' 2  —  .-r'3 
-7-7 rrrX^  —  ic').     For  the  point  on  the  curve  ic  =  2,  this  gives  for  the  two 

y  (.*    "^ ) 

tangents  y  z=l'^x  —  2,  and  y  =  —  2a;  +  2. 


200,  CoE. — The  tangent  of  the  angle  v)hich  a  tangent  to  an  ellipse  at 

B^x  B-x 

(x,  y)  makes  with  the  axis  of  abscissas  is r — ,  to  an  hyperbola ,  and 

A-y  -"^  A2/ 

to  a  parabola  -. 

y 

y'  ifj" 

Dem.— These  are  the  values  of  the  coefficient  —, •—  as  obtained  in  the  several 

cases  in  Ex.  4,  10,  and  1.     But  this  coefficient  is  the  tangent  of  the  angle  which  the 
hne  makes  with  the  axis  of  x  {31,  Cob,  1). 

Ex.  1.  What  angle  does  a  tangent  to  y-  ==  8x  make  with  the  axis  of 
X,  when  the  point  of  tangency  is  at  ^  =  2  ?     At  ^  =  5  ?     At  ^  =  10  ? 
Answers,  Tan"^!  or  45°,  tan-^— 1)  or  135°  ;   tan-^(  ±  .63245)  or 

32°  18'  30"  and  147°  41'  30";    tan"^  (=b  .447214)  or  24°  5'  41" 

and  155°  54'  19". 

Ex.  2.  What  angle  does  the  focal  tangent  to  the  parabola  make 
with  the  axis  of  ^  ? 

Ex.  3.  In  an  ellipse  whose  axes  are  10  and  6  what  angle  does  a 
tangent  at  the  point  x  =  2  make  with  the  axis  of  ^(7  ? 

Ans.,  165°  19'  33"  and  14°  40'  27". 

270*  ScH.  — To  determine  at  what  point  on  a  given  curve  a  tangent  must 
be  drawn  to  make  a  given  angle  with  the  axis  of  x ;  or,  what  is  the  same 
thing,  to  find  a  point  at  which  a  curve  has  a  given  direction,  put  the  gen- 
eral value  of  the  co-efficient  of  x  in  the  equation  of  the  tangent  to  the  par- 
ticular curve  equal  to  the  tangent  of  the  proposed  angle.  The  equation 
thus  formed  together  with  the  equation  of  the  locus  will  enable  us  to  find  the 
values  of  x  and  y  which  locate  the  point  sought.  In  case  the  tangent  is  to  bo 
parallel,  the  co-efficient  is  put  equal  to  0  ;  if  perpendicular,  equal  to  oo. 

Ex.  4.  At  what  point  on  an  ellipse  whose  axes  are  16  and  8  must  a 
tangent  be  drawn  to  make  an  angle  with  the  axis  of  x  whose  tangent 
is  2  ?  At  what  point  to  make  an  angle  of  45°  ?  At  what  point  to 
make  an  angle  of  135°  ?  At  what  point  is  the  tangent  parallel?  At 
what  point  perpendicular  ? 

Sug's. — The  general  value  of  the  tangent  of  the  angle  which  a  tangent  to  an 

B^x 
ellipse  makes  with  the  axis  of  x  is  —  - — .     Hence  for  the  1st  inquiry  we  have 

16a; 
--  ^-  =  2,  and  Gl?/-  +  16.r-2  ==  1024,  from  M^hich  to  find  a;  and  y. 


SUBTANGENTS. 


179 


Ex.  5.  At  what  point  on  an  hyperbola  whose  axes  are  8  and  6  must 
a  tangent  be  drawn  to  make  an  angle  of  45°  with  the  axis  of  ^? 
"What  angle  does  the  focal  tangent  make  with  the  axis  of  ^  ? 

Ex.  6.  On  an  elliptical  track  whose  transverse  axis  is  1  mile,  and 
whose  conjugate  is  f  of  a  mile  in  length  and  coincides  with  the  me- 
ridian, in  what  direction  is  a  man  travelling  when  he  is  on  the  north- 
west quarter  of  the  track,  travelling  around  from  left  to  right,  and  is 
at  40  rods  from  the  transverse  axis  ? 

Answer,  North  25°  14'  21"  east. 


SUBTANGENTS. 

271*  r>EF. — A  Subtangent  is  the  portion  of  the  axis 
scissas  intercepted  between  the  foot  of  the  ordi- 
nate from  the  point  of  tangency,  and  the  inter- 
section of  the  tangent  with  this  axis  ;  or  it  may 
be  defined  as  the  projection  of  the  correspond- 
ing portion  of  the  tangent  upon  the  axis  of  x. 
DT  is  the  subtangent  corresponding  to  the 
point  p. 


of  ab- 


272.  ^Toh, — To  find  the  length  of  the  sub- 
tangent. 


Fig.  188. 


Solution. — Letting  a  represent  the  angle  which  a  tangent  to  the  curve  at  the 
specified  point  makes  with  the  axis  of  x,  find  tan  a  as  in  {268).     Whence  from  the 

PD       ,  ,,,         ,  ,  ,.  y 


triangle  PDT",  Fig.  188,  we  have 


tan  a:,  or  D  T  (the  subtangent) 


DT  ^  "'tana: 

A  slightly  diff'erent  method  of  solution  is  to  produce  the  equation  of  the  tangent 
to  the  curve,  and  then  find  where  it  intersects  the  axis  of  x.  This  intercept,  as 
AX,  Fig.  188,  and  the  abscissa  AD  make  known  the  subtangent,  which  is  always 
their  algebraic  difference. 

Ex.  1.  Eind  the  value  of  the  subtangent  of  the  common  parabola 
at  {x',  y'). 


Solution. 
«/'2       2px' 


-We  find  in  {268  Ex.  1)  that  tan  a  =  ^,.     Whence  Suht 


y 


y 


tana 


V 


=  -^-  =  2a;'. 


273,  ScH. — ^From  this  example  we  learn  that  the  subtangent  in  a  para- 
bola is  always  equal  to  twice  the  abscissa  of  the  point  of  tangency.  Hence 
to  draw  a  tangent  to  a  parabola  at  any  point  as  R,  Fig.  188,  let  fall  the  ordi- 
nate PD,  take  AT  ==  AD,  and  draw  PT. 


180 


PBOPERTIES   OF   PLANE   LOCI. 


Ex.  2.  What  is  tlie  subtangent  to  y^- :=  10^,  at  .r  =  6?     At2/===8? 
At  2/  =  12  ?     Draw  these  tangents  according  to  the  schohum. 


Ex.  3.  Find  the  value  of  the  subtangent  to  the  ellipse  at  {x',  y'). 


SoLTTTioN. — "We  have  tana 
A^y'-i       A^  —  .r'2 


.     ,hjEx.4.  {268),  yrhenceSuht.  =  ^— 
-A'^y  tan  a 


B-x  x 

not  a  direction. 


neglecting  the  —  sign  as  we  are  inquiring  simply  for  a  value, 


Fig.  189. 


274:,  ScH.  1. — From  this  example  we 

learn  that  the  subtangent  in  the  ellipse 

does  not   depend  upon  the  conjugate 

axis,   but  only  on  the   transverse  axis 

and  the  abscissa  of   the  point  of  tan-   C| 

gency.      Hence    if  sevei^al    ellipses    he 

drawn  on  the  same   transverse   axis  the 

subtangenis   corresponding    to    the   same 

abscissa  are  equal.     Thus  in  Fig.    189 

let  APB,  AP'B,  AP'  B,  and  AP"B  be 

several  ellipses  on  the  same  transverse  axis  CB,  then  AD  being  any  abscissa 

[x'),  the  subtangent  corresponding  to  each  point  of  tangency  P,  P',  P",  P'",  is 

J^i  x'- 

'- — .     Hence  the  tangents  at  these  several  points  cut  the  axis  of  x  at 

x 

the  same  point  T. 

275,  ScH.  2. — This  property  affords  a  convenient  method  of  drawing  a 
tangent  to  an  ellipse  at  a  given  point  in  the  curve.  Thus  let  P  be  the 
point,  Fig.  189.  Draw  the  circle  upon  the  transverse  axis  CB,  draw  the 
ordinate  PD  and  produce  it  till  it  meets  the  circumference  of  the  circle 
in  P",  and  then  draw  a  tangent  to  the  circle  at  P'".  T  being  the  point 
at  which  the  latter  intersects  the  axis  of  x,  is  also  the  point  at  which  au 
corresponding  tangents  to  ellipses  on  the  same  axis  intersect.  Hence  draw- 
ing PT  it  is  the  tangent  sought. 

276,  CoK.  — The  expression  for  the  subtangent  being  independent  of 
B,  the  property  is  the  same  in  the  hyperbola  as  in  the  ellipse  {00).     How- 

x'2  —  A2 


ever  as  x  ^  A  in  the  hyperbola,  we  may  write  Subt. 
to  have  it  positive. 


in  order 


Ex.  4.  I^rojy* — In  an  ellipse  or  hyperbola  if  from  any  point  in 
the  curve  a  tangent  and  an  ordinate  be  drawn  to  the  transverse  axis, 
half  this  axis  is  a  mean  proportional  between  the  distances  of  the 
intersections  from  the  centre.. 


NOEMALS. 


181 


Dem. — Let  P  be  any  point  in  tlie  curve  and 
PD,  PX  the  ordinate  and  tangent.  Then 
AX  :AB  ::AB:AD.  For  we  have  the 
equation  of  a  tangent  at  P  {x',  y'),  A''yy'  rt 
B'^xx    =  ±  A^B\      Making  ?/  =  0,    we   get 

x=—,.     But  a  is  AX  and  x  is  A D.     Hence 

X 

X  :  A  ::  A  -.x,  or  AX  :  A B   :  :  A B  :  A D. 

The  demonstration  being  the  same  in  each  case. 


Fig.  190. 


Fig.  191. 


277*  ScH  1. — To  drav)  a  tangent  to  a 
hyperbola  at  a  giveri  point.  From  the 
given  point  of  tangency  P,  F'ig.  191,  let 
fall  the  ordinate  PD  ;  and  upon  the 
transverse  axis  HB,  and  the  abscissa  AD, 
draw  semi-circumferences.  From  their 
intersection  let  fall  LT  a  perpendicular 
upon  the  axis  of  x.  Draw  a  line  through 
P  and  T  and  it  is  the  tangent  sought. 
Proof.  Drawing  AL  and  LD,   we  have  AD   {ov  x')   :  AL   (or  Jl)   : :  AL 

(or  A)  :  AT.     Whence  AT  =   —  and  is  the  intercept  made  by  a  tangent 

at  p. 

278,  ScH.  2.— The  pupil  can  scarcely  fail  to  notice  the  close  analogy 
between  the  forms  of  the  equations  of  the  conic  sections  and  the  equations 
of  their  tangents ;  6y  simply  dropjnng  the  accents  the  latter  return  to  the 
former.     Thus  dropping  the  accents 

A^-yy'  +  B^xx'  =  A'^B'- becomes  A^y'^  -f  B'^x'^  =  A^B% 

^'^yy'  —  B-^xx'  =  —  A^B^ becomes  A^y^  —  B^x"^  =  —  A^B% 

yy'  =  P{^  +  ^') becomes  y^  =  2px,     and 

yy'  -^  xx'  =z  E-2 becomes  y^  -{-  x^  =  Rk 


NORMALS. 

279,  Dep. — A  l^OTTYial  to  a  plane  curve  is  a  perpendicular  to  a 
tangent  at  the  point  of  tangency. 

280,  JPvoh, — To  produce  the  equation  of  a  normal  to  a  plane  curve. 

Dem. — Let  PE  be  a  normal  to  the  curve 
M  N  at  the  point  P,  the  co-ordinates  of  which 
are  {x\  y').  The  equation  of  a  tangent  at  the 
point  P  is  2/  —  y'  =  a{x  —  a;'),  in  which  a  is 
the  tangent  of  the  angle  which  the  tangent  at 
P  makes  with  the  axis  of  x,  that  is  tan  SXX  ; 
and  is  to  be  determined  from  the  equation  of 
the  curve  as  in  the  preceding  part  of  this  sec- 
tion.     Now    the    equation  of   a  line    passing 


Fig.  192. 


182  PKOPERTIES  OP  PLANE  LOCI. 

tkrougli  (x'j  y')  and  perpendicular  to  the  line  y  —  y'  =  cl{x  —  x),  is  y  —  y'  =^ 
' {x  —  a;')  {39),  the  coefficient being  the  negative  reciprocal  of  the  tan- 
gent of  the  angle  which  the  tangent  to  the  curve  makes  with  the  axis  of  x. 

251,  CoE. — The  tangent  of  the  angle  luhich  a  normal  to  a  curve  at  a 
particular  point  makes  ivith  the  axis  of  x  is  the  negative  reciprocal  of  the 
mngent  of  the  angle  which  is  made  by  a  tangent  to  the  curve  at  the  same 
point. 

Ex.  1.  To  find  the  equation  of  a  normal  to  an  ellipse. 

Solution. — The  equation  of  the  tangent  to  the  ellipse  is  A-yy'  -f-  -B-xx'  =  A^B^, 

B^x'         B- 

ov  y  = J7~r'^  H r-     Now  the  equation  of  any  Hne  passing  through  {x,  y')  is 

A-y         y 

y  —  y'  =  ci{x  —  x')  ;  but  in  order  that  this  should  be  a  normal  to  the  ellipse  we 

must  have  a  =  ,,^'  ,,  the  negative  reciprocal  of  the  tangent  of  the  angle  which  the 

A'^y' 
tangent  makes  with  the  axis  of  x.     Hence  y  —  V'  =  -^^—X^  —  ^')  is  the  equation 

JD-X 

of  a  normal  to  the  ellipse. 

Ex.  2.   Show  that   the  equation  of  a  normal  to  an  hyperbola  is 

Ex.  3.   Show  that   the   equation  of    a  normal  to  the  parabola  is 

I        y'  f       i\ 

y  —  y' =  ~ -{oc  —  x'). 

y' 
Ex.  4.  Show  that  the  equation  of  a  normal  to  the  circle  is  2/  ==  —,oc^ 

and  hence  is  the  radius. 

Ex.  5.  What  is  the  equation  of  a  normal  to  the  elhpse  whose  axes 
are  8  and  4  at  ^  =  1?     At  a:  =  —  1?     At  ^  =  3? 

One  equation  is  y  =  ^  ^vlx  qp  f  v  7. 

Ex.  6.  What  is  the  equation  of  a  normal  to  the  parabola  whose 
parameter  is  9,  at  a:  =  4  ?     At  ^  =  9  ?     At  ^  =  5|-  ? 

One  equation  is  1/  =  qp  2^  db  27. 

252,  Cor. — The  general  expressions  for  the  tangents  of  the  angles 

which  normals  to  the  conic  sections  make  with  the  axis  of  x  are  : 

A-y'  A^y' 

For  the  ellipse  -^7^,  for  the  hyperbola  —  :^ — ,, 

y'  Y 

For  the  circle        — ^,  for  the  parabola        —  — . 


SUBNOBMALS. 


183 


SUBNORMALS. 

283,  Def. — The  Subnormal  is  the  projection  of  the  normal 
upon  the  axis  of  x  ;  or  it  is  the  distance  from  the  foot  of  the  ordinate 
let  fall  from  the  point  in  the  curve  to  which  the  normal  is  drawn,  to 
the  intersection  of  the  normal  with  the  axis  of  ^,  as  D  E,  Fig.  193. 

Ex.  1.  Show  that  the  subnormal  in  the 

B'^x' 
ellipse  is  equal  to 


A^' 


and  has  the  same 


Whence 


numerical  value  in  the  hyperbola. 

Sug's. — In  case  of  the  elhpse,  let  PE  be  the 
normal  at   P   and   ED    the   subnormal.      Now 

--— -  =  tan  PE  D,  or  -4 —  =  —^,. 
E  D  Subnor       B'^x 

Subnor  =  — —. 
A'' 

Ex.  2.  Show  that  in  the  parabola  the  subnormal  is  constant  and 
equals  the  semi-parameter.  Show  how  this  property  may  be  used  to 
draw  a  tangent  to  a  parabola  when  the  focus  is  known. 


Fig.  193. 


284.  JPvop, — In  the  parabola  a  line  joining  the  focus  and  the  inter- 
section of  a  tangent  with  the  axis  of  j  {a  tangent  at  the  vertex),  is  per- 
pendicular to  the  tangent. 

Dem. — Let  F  be  the  focus,  PT  a  tangent,  and 
PE  a  normal.  Join  the  intersection  L  with  F. 
Then  is  FL  perpendicular  to  PX.  For,  since 
AT  =  AD  (^.  1,  272),  TL  =  LP.  Again 
TF  =  AX  +  AF  =  AD  +  ^p.  Also  FE  = 
AD+DE  — AF  =  AD+p  — ip  =  AD  +  ip. 
"Whence  as  X  P  and  X  E  are  bisected  by  F  L  it  is 
parallel  to  the  normal  P  E  and  hence  perpendicular 
to  the  tangent  PX.     Q.  e.  d. 

Fig.  194. 

28d.  Cor.  PF=TF=FE.  Also  angle  PT  F  =  T  P  F,  and 
FPE=FEP.    ^^am  PFX==2PTX. 

286.  ScH.  1. — Having  given  the  curve  and  its  axis,  to  find  the  focus 
draw  a  tangent  to  any  point,  as  P,  by  (273),  and  then  erect  a  perpendicular 
to  it  where  it  intersects  the  tangent  at  the  vertex.  The  intersection  of  this 
perpendicular  with  the  axis  of  x,  will  be  the  focus 

287-  ScH.  2. — Having  given  the  axis  and  focus,  to  draw  a  tangent  and  a 
normal  at  P,  take  FX=  FP=  FE  and  draw  PX  and  PE. 

288,  ScH.  3.— Having  the  axis  and  focus,  a  tangent  may  be  drawn  making 


184 


PROPERTIES   OF   PLANE   LOCI. 


any  given  angle  with  the  axis  of  x,  by  making  P  FX  =  twice  the  given  angle, 
and  drawing  a  tangent  from  the  point  where  P  F  intersects  the  curve.. 


^  »  » 


EJECTION  XV, 
Special  Properties  of  tlie  Oonio  Sections. 

[Note. — The  impoi-tance  of  the  Conic  Sections  renders  it  necessary  that  their  properties  should 
be  more  fully  developed  than  is  found  expedient  in  a  compendious  presentation  of  the  subject  of 
the  General  Geometry,  and  hence  this  section.  Similar  sections  might  be  added  on  other  curves, 
as  of  the  cycloid,  the  catenary  ;  or  sections  discussing  the  loci  embraced  by  equations  of  the  3rd 
degree,  or  the  4th  degree,  etc.  But  these  subjects  are  not  of  sufficient  importance  to  reqinre 
treatment  in  an  elementary  course,  nor  capable  of  being  epitomized  so  as  to  be  brought  within 
proper  hmits  for  such  a  course.  Those  who  wish  to  pursiie  the  subject  farther  wUl  find  Salmon's 
Conic  Sections  and  Higher  Plane  Curves  in  two  volumes,  or  Price's  Infinitesimal  Calculus  in 
four  large  volumes,  the  best  English  resources.  Todhunter's  four  volumes,  two  on  the  Calculus, 
and  two  on  the  Co-ordinate  Geometry,  are  also  among  the  most  valuable  recent  treatises.  The 
author  of  this  volume  proposes  to  prepare  a  second  volume  on  loci  in  space,  and  a  more  extended 
course  in  the  Calculus,  as  soon  as  he  is  able.] 


(a)  EADII  VECTOEES  AND  THE  ANGLES  WHICH  THEY  MAKE  WITH 

A  TANGENT. 

2S9,  Def. — A  Radius  VectOi^  of  a  conic  section  is  a  line 
drawn  from  a  focus  to  a  point  in  the  curve. 

200,  JPTop, — In  an  ellipse  the  sum  of  the  radii  vectores  to  any 
point  in  the  curve  is  constant  and  equal  to  the  transverse  axis,  and  in  the 
hyperbola  the  difference  is  constant  and  equal  to  the  transverse  axis. 

Dem.      P  being  any  point  in  the  curve,      M 
let  PF  =  r  and  PF'  =  r' .     We  have 


V  P  D"'-f  D  f' =  v/^HM^e" 


XV-' 


Fig.  195. 


also  r'=V  p  D^-f  D  F''^=^Vy'-\-{Ae-\-xf. 
But  2/2  =  (^2  _  a;2)(i  _  e2)  {S2),  Sub- 
stituting this  value  of  y-,  we  have 
r  =  \/[A^  —  x^){i  —  e'^)  -f  {Ae  —  x)-^  = 
\/A^  —  2Aex  -\-  e^x^  =  A  —  ex\  and 
r'  =  V{A^—x^}{l  —  e'^)-\-[Ae  -f-  03)2  =  v/^a  _|_  ^Aex  +  e^'^  =  A-\-  ex.  Adding, 
r'  -\-  r  =.  2A,  for  the  ellipse. 

In  the  hyperbola  A  —  ex,  is  negative,  since  ex  ^  A,  and  we  write  j-'  =  JL  -f-  ex, 
and  r  =  ex  —  A.     Whence,  subtracting,  r'  —  r  =  2  J.,     q.  e.  d. 

201,  CoR.  —  The  length  of  a  radius  vector  drawn  to  the  nearer  focus 
t.s  r  =  A  —  ex,  and  to  the  more  remote  x'  =  A  -f  ^x. 


SPECIAL   PKOPERTIES    OF   THE    CONIC    SECTIONS. 


185 


N 


•  292,  ScH. — The  principles  enunciated  in  tliis  proposition  afford  very 
simple  means  for  constructing  the  loci  mechanically.  For  the  ellipse  take 
a  string  equal  in  length  to  the  transverse  axis,  and  fastening  its  ends  at  the 
foci,  put  a  pencil  against  the  string  and  move  it  around  the  perimeter  of 
the  cm-ve,  keeping  the  string  tense. 
Thus  F'PF  Fig.  195,  represents  the 
string,  and  P  the  pencil  when  the 
point  P  is  located. 

To  construct  an  hyperbola,  take  a 
ruler  AB,  and  a  string  BPF  ;  mak- 
ing the  string  shorter  than  the  ruler 
by  the  length  of  the  transverse  axis 
of  the  required  hyperbola ;  fasten 
one  end  of  the  string  to  one  end  of 
the  ruler,  as  at  B,  fasten  the  other  end  of  the  string  at  one  focus,  as  at  F,  and 
the  other  end  of  the  ruler  at  the  other  focus,  as  at  F'.  Place  a  pencil 
against  the  string  and  bear  it  against  the  side  of  the  ruler,  as  at  P,  and 
keeping  the  string  tense  move  the  pencil  around  the  curve.  It  is  evident 
that  F'P  —  PF  =  2 A  in  all  positions  of  P.  Hence  P  traces  the  curve. 
To  trace  the  other  branch  the  attachments  have  to  be  changed,  so  that 
the  free  end  of  the  string  shall  be  attached  at  F',  and  the  end  of  the 
ruler  at  F. 


Fia.  196. 


203,  JPtop* — The  radii  vectores  drawn  to  any  point  in  an  ellipse  of 
hyperbola  make  equal  angles  with  the  tangent  at  that  point. 

Dem. — Let  PF  and  PF',  be  the  radii 
vectores,  and  M  T  the  tangent,  Fig's.  195, 
197.  ThenFPT=F  PM.  For, AT  = 

—  {137,  Bx.  1,  ox  276,  Ex.  4),  and  AF 

=:A  F'==Ae.  Hence,  in  the  ellipse,  F"r= 

A  A 

~{A  —  ea'),   and   F'X  :=  —{A  4-  ex)'. 

X  X 

and  in  the  hyperbola  FT  =  —  (ex  — A)^ 


and  F'T 


-(ex  -f  A),     Wherefore  in 


Fig.  197. 


either  case  we  have  FT  :  F'T  : :  r  :  r'  {201).  Now  drawing  F'M  parallel  to 
PF  we  have  FT  :  FT  : :  PF  :  F'M,  or  r  :  r'  : :  r  :  F'M.  .• .  F'M=r'  = 
F'P,  and  F  MP  =  FPT=  F'PM.     q.  e.  d. 

204:,  CoE. — In  the  ellipse  the  normal  bisects  the  angle  included  by  the 
radii  vectores  to  the  same  point ;  and  in  the  hyperbola  it  bisects  the  angle 
included  by  one  radius  vector  and  the  other  produced. 


186 


PKOPEKTIES   OF  PLANE  LOCI. 


ivi; 

Fig.  198. 


295,  ScH.— This  principle  affords  one  of  the  most  convenient  methods 
Df  drawing  tangents  to  these  curves. 

1st.  To  draw  a  tangent  through  a  giveji  point  in  the  curve.  Let  P,  Fig's. 
195,  197,  be  the  point.  Draw  the  radii  vectores  to  the  point  and  bisect  the 
included  angle  for  the  hyperbola,  or  the  angle  included  by  one  radius  vector 
and  the  other  produced  in  the  case  of  the  elHpse. 

2nd,    To  draw  a  tangent  frovx  a  point  with- 
out the  curve.     Let  P  be  the  point.     Join 
P  with  the  nearer  focus,  and  from   P  as  a 
centre  pass  an  arc  of  a  circle  through  that 
focus.     From  the  other  focus,  with  a  radius 
equal  to  the  transverse  axis,  strike  an  arc 
cutting  the  former  as  at   D  and  D'.     Join 
D  and  D'  with  F',  and  T  and  T'  are  the 
points  of  tangency.     To  prove  this  for  T, 
join  D  and  F,  and  F  and  T.     Now  F'T  + 
TF  =:  F'D,  since  each  =2-4.     Hence 
TD  =  TF.      Moreover    DP  =  PF. 
Hence  "TP  is  perpendicular  to  DF,  and 
angle  DTP  =  MTF'=PTF.  Whence 
we  know  that  PM  is  tangent  at  T.    In 
a  similar  manner  PM'  can  be  shown 
perpendicular  to  FD',  and  hence  tan- 
gent at  T'.     [The  student  should  com- 
plete the  figure  and  give  the  demon- 
stration in  the  case  of  the  hyperbola.]  Fig.  199. 

3rd.  If  a  circle  be  described  on  PF  and  another  on  CB,  the  lines  passing 
from  P  through  their  intersections  are  tangents  to  the  curve,  and  this 
whether  P  is  in  or  without  the  curve.  [Why?  The  student  should  be 
able  to  answer  after  having  read  in  this  section  through  the  next  two  propo- 
sitions.] This  method,  however,  is  imjDracticable  when  p  is  without  the 
curve,  as  it  does  not  indicate  the  precise  point  of  tangency. 


200,  JProp, — In  the  'parabola  the  radius  vector  drawn  to  the  point 
of  tangency  makes  the  same  angle  with  the 
tangent  as  a  diameter    through   the  same 
point,  or  as  the  tangent  does  with  the  axis  of 
abscissas. 


Dem. — Let  PF  be  the  radius  vector,  PF'  the 
diameter,  and   MT  the  tangent.     Then  FPT 
=  F'PM.   For,  AT=  X  {140,  ov  272,  Ex.  1), 
and  A  F  =  -^p.  Hence  FT  =  x  -f  Ip.  But  PF 
=  PD  =  BE=  BA  + AE=rJp-f-a;.     .-.FT 
=  FP,  and  angle  FPT=:  FTP=F'PM.  q.e.d. 


Fig.  200. 


SPECIAL  PBOPERTIES  OF  THE  CONIC  SECTIONS. 


187 


297*  Cor. — In  the  parabola  the  normal  bisects  the  angle  included  by 
the  radius  vector  and  a  diameter  at  the  same  point  in  the  curve. 

298,  Sen.— To  draw  a  iangerit  to  the  poiiit  P  in  the  parabola,  draw  the 
radius  vector  and  the  diameter  to  the  point,  produce  one  of  them  (as  PD) 
and  bisect  the  angle  thus  formed. 

To  draw  a  tangent  from  a  point  without  as  P",  join  the  point  with  the 
focus,  from  P"  as  a  centre,  pass  an  arc  of  a  circle  through  the  focus,  and 
through  its  intersections  with  the  directrix  draw  diameters.  The  vertices 
of  these  diameters  are  the  points  of  tangency  on  the  curve,  as  P  and  P'. 
To  prove  this,  observe  that  as  PF  ==  PD,  and  P"F  =  P"D,  P"P  is  per- 
pendicular to  DP  and  angle  FPP"  =  DPP"=  MPF'.  [Let  the  stu- 
dent give  the  proof  for  the  point  P'.] 

[Note.— The  properties  demonstrated  in  these  propositions  give  elliptic,  hyperbolic,  and 
paraboUc  reflectors  their  pecuUar  properties.  Thus,  rays  of  light,  sound,  or  heat  diverging  from 
one  focus  of  an  elliptic  reflector  aie  converged  at  the  other  ;  diverging  from  one  focus  of  an 
hyperbolic  reflector,  they  diverge  after  reflection  as  though  they  proceeded  directly  from  the 
other  focus  ;  and  diverging  from  the  focus  of  a  parabolic  reflector,  they  are  reflected  parallel. 
Conversely  to  the  last,  incident  rays  parallel  to  the  axis  are  concentrated  at  the  focus  of  a  para- 
bolic reflector.] 


200,  JPvop, — The  rectangle  of  the  perpendiculars  from  the  foci 
upon  the  tangent  of  the  ellipse  or  hyperbola  is  constant,  and  equal  to  the 
square  of  the  semi-conjugate  axis. 

Dem. — Let  L'X  be  a  tangent  at  any  point  P,  and 
FL  a  perpendicular  from  the  focus  upon  it.     Pro- 
duce FL  till  it  meets  F'P,  produced  if  necessary, 
in  D,  and  draw  AL.     Since  AF  =  AF'  and  FL 
=  LD,  A  L  is  parallel  to  F'  D  and  equal  to  i  F'  D 
=  A  B.     Hence  the  foot  of  a  perpendicular  from  the 
focus  upon  a  tangent  lies  in  the  circumference  of  a  circle 
described  on  the  transverse  axis.     Now  let  F'L'  be 
the  perpendicular  from  the   second  focus  upon  the 
tangent  L  T,  we  are  to  show  that  FL  X  F'L'  = 
BK     Join   A  and   L'  and  produce  the  line 
till  it  meets    LF   produced  in  L".     Then 
the  triangles  AL'F'  and  AL"F  are  equal, 
and  AL"  =  AL',  and  L"  is  in  the  cir- 
cumference of  the  circle  described  on  C  B. 
Finally,    FL  X   F'L'  =  FL  X    FL"  == 
FB  X   FC  (CB  and  LL"  being  chords 
in  the  same  circle).      But    FB  X   FC  == 
{A  -f-  Ae){A  —  Ae)  =  ^^(l  —  e^)  =  B^  {49, 
3rd  and  7th).        .-.   FL   X    F'L'   =  B\ 

Q.  E.  D. 


Fig.  202. 


300.  Cor. —  The  semi-conjugate  axis  is  a  mean  proportional  between 
Qui  focal  distances. 


188 


PKOPERTIES    OF   PLANE   LOCI. 


(&)    SUPPLEMENTAHY  CHOKDS  AND   CONJUGATE  DIAMETERS. 

801.  Def. — The  term  Ordinate,  as  used  in  connection  with  the 
conic  sections,  may  mean  any  line  drawn  from  a  point  in  the  curve  to 
any  diameter,  and  parallel  to  a  tangent^  at  the  extremity  of  that 
diameter. 

302.  Def. — SupplemeTitary  Chords  are  chords  drawn 
from  any  point  in  the  curve  to  the  extremities  of  any  diameter. 

303,  Def. — One  diameter  is  said  to  be  conjugate  to  another 
when  it  is  parallel  to  a  tangent  at  the  extremity  of  the  latter. 


304:,  JProp, — In  an  ellipse  the  rectangle  of  the  tangents  of  the  angles 
which  a  pair  of  supplementary  chords  make  with  the  transverse  axis  is 
B2 


equal  to 


A2' 


Dem, — Let  PC  and  PB  be  supplemen- 
taiy  chords  drawn  to  the  axis.  Let  the 
angles  PCX  and  PBX  be  represented  by 
a  and  a',  and  tan  a  =  a,  and  tan  a  =  a'. 

Then  aa'  =  — :  -— ,  A  and  B  being  the  semi- 

axes.     The  equation  of  PC  isy=a{x-\-A), 

and  of  PB  y  =  a'{x  —  A),   disregarding 

the  signs  of  a  and  a',  since  PC  and  PB 

are,  as  yet,  any  lines  passing  through  C,  and  B.     For  the  intersection  of  thest 

lines  these  equations  are  simultaneous,  and  we  may  have  y"^  =  aa!  {pfl  —  A^^. 

Again,  when  the  point  of  intersection  is  in  the  ellipse,  this  result  is  simultaneous 

with  the  equation  of  the  curve,  A-yl^  -\-  B-x-  =  A'^^B'^,  ox  y^= —{pfi  —  A~)  ; 


D 

Fig.  303. 


A^^ 


B-i 
A- 


whence  combining  the  two,  we  have  aa'=:  —  -— .     q.  e.  d. 


303,  CoK.  1. — By  a  similar  course 
of  reasoning,  or  by  simply  changing 
the  sign  of  B^*,  we  have  for  the  hyper- 

B^ 

I? 


bola  aa' 


306,    CoK.  2. — In  the    ellipse,  if 

supplementary  chords  are  drawn  to  the 

extremities  of  the  conjugate  axis  aa'= 

B2 


Fig.  204. 


-j—  {the  same  as  before)  if  the  angles  are  measured  from  the  axis  of  x, 

A^ 


hut  aa'  =  — 


A^ 


—  if  they  are  measured  from  the  axis  of  y. 


SPECIAL   PROPERTIES   OF   THE   COXIC    SECTIONS. 


189 


Dem. — The  equations  of  P'E  and  P'D,  Fig.  203,  are  respectively  y  —  B  =  ax 
and  y  -j--  B  =  a' X  ;  whence  t/^  —  B^  =  aa'x'^.     From  A-y^  -j-  B-x^  -.=  A^B^  we  have 

^2  B^ 

2/2  —  B-  = —x'^.     . ' .  aa'  = ■— .     If  the  angles  are  reckoned  from  the  axis 


A^ 


A^ 


of  y  we  observe  that  for  a  we  shall  have ,  and  for 


,  1 

a  ,  -. 
a 


aa  = 


4! 


307 •  Cor.  3. — In  the  hyperbola,  if  supplementary  chords  are  drawn 
from  any  point  in  the  conjugate  curve  to  the  extremities  of  the  conjugate 

axis  aa'  =.  —  if  the  angles  are  reckoned  from  the  axis  of  x,  hut  aa'  = 


A2  . 


B2 


if  they  are  reckoned  from  the  axis  of  j. 


Dem. — [Let  the  student  supply  the  demonstration,  and  also  show  that  if  the 
chords  be  drawn  from  a  point  in  the  x  hyperbola  to  the  extremities  of  the  conju- 
gate axis,  aa'  is  not  constant.  ] 

SOS*  Cor.  4. — In  the  circle  this  relation  becomes  aa'  =  —  1,  or 
1  -}-  aa'  =  0  ;  which  shows  that  the  chords  are  perpendicular  to  each 
other,  which  is  a  well  known  property  of  the  circle.  In  the  equilateral  hy- 
perbola the  relation  is  aa'  =  1,  or  a  ^=  — ,  signifying  that  the  angles  are 

a 

complementary. 

309,  Cor.  5. — ^  one  of  two  supplementary  chords  to  either  axis  is 
parallel  to  one  of  two  drawn  to  the  other  axis  the  other  two  are  parallel. 


.  If  a  is  the 


Dem.— In  either  Fig.  203  or  204  if  P'E  is  parallel  to  PC,  P'D  is  parallel  to 

PB.     For  we  have  m  case  of  each  set  of  chords  aa  =  -t-  ~— . 

A'^ 

same  in  each,  a'  is  also. 

310.  ScH. — The  —  sign  in  the  formula 

indicates  that  a  and  a'  have  op- 


aa  = 


A-^ 


posite  signs  in  the  ellipse.     Thus  P  being 

the  point  from  which  the  chords  are  drawn, 

PBX  >  90°-  and  <  180°  gives  —  a,  PCX 

<C  90°,  gives  +  a.     Again  P"BX  being  an  ^i*^-  205. 

angle  between  180°  and  270°,  tan  P"  BX  =  +  a',  but  P"CX  being  between 

270°  and  360°,  its  tangent  is  —  a.     In  a  similar  manner  the  +  sign  in  the 

formula  aa'  =  -\ signifies  that  a  and  a'  have  always  the  same  sign,  aS 

may  be  readily  observed  from  a  figure. 


311.  I^Toh, — To  discuss  the  angle  included  between  supplementary 
chords  to  the  transverse  axis  of  an  ellipse. 


190  PROPERTIES  OF  PLANE  LOOT. 

SoiiUnoN. — Let  V  be  the  included  angle  CPB  ;  then  tan  V  =  - — : = 

1  4-aa 

-J-.    Limiting  the  discussion  to  the  upper  segment  CPB,  Fig.  205,  a'  is 

always  — .     Hence  tan  V  is  — ,  and  V  is  always  an  obtuse  angle. 
♦Again,  as  V  is  a  variable  angle,  we  may  inquire  when  it  is  a  maximum.     Differ- 

.    .  B^ 

entiatmg  a    -\ — —-;  with  respect  to  a',  and  putting  the  result  =  0,  we  have 

a  =  ±  —.     The  ambiguous  sign  is  explained  by  the  fact  that  the  result  applies 

equally  well  to  either  angle  PBX,  or   PCX.     Hence  V  is  a  maximum  when 

7?  7? 

tan  P  BX  = 7-,  or  tan  PCX  ==  — ,  L  e.,  when  P  is  at  D. 

A  A 

To  find  the  value  of  V  when  it  is  a  maximum,  we  have  simply  to  substitute 
;;  for  a'  in  tan  V  ;  whence  tan  V  = 


A  '  A^  —  B^' 

ScH. — The  angle  included  by  supplementary  chords  to  the  conjiligate 
axis  of  an  ellipse,  and  also  the  corresponding  cases  in  the  hyperbola,  may 
be  discussed  in  a  similar  manner ;  but  the  results  are  not  important. 


312»  JPvop, — ^  one  of  two  supplementary  chords  to  the  transverse 
axis  of  an  ellipse  is  parallel  to  a  tangent,  the  other  is  parallel  to  the  diam- 
eter drawn  through  the  point  of  tangency,  and  conversely. 

Dem.  — Let  P  B  be  parallel  to  M  "T,  then  is 
PC  parallel  to  DD'.  -Let  tan  PCX  =  a, 
tan  PBX  =  a,  tan  DAX  =  a^,  and 
tan  M  TX  =  a, '.     Now  the  equation  of  D  D' 

is  2/  =  diX  ;  whence  «,==-.     Also  tan  MXX     /- 

=  _  ?^^136,Ex,.l,  oi 269).Rence,  aia,'=.-~.D 
A'^y  A- 

B- 

But  aa'  = --.     .• .  aa'  =a,a.\  and  if  a'  =  Fig.  206. 

A^  11' 

a/,  a  =  «!•     Conversely,  if  a  =  a-^,  a'  =  a-^ .     q.  e.  d. 

313»  Cor.  1. — The  same  property  exists  in  the  hyperbola  and  is  dem- 
onstrated in  the  same  way. 

314,  ScH.— This  property  affords  a  convenient  method  of  drawing  tan- 
gents. Thus  to  draw  a  tangent  at  D  Fig's.  206,  207,  draw  the  diameter  D  D', 
the  chord  PC  parallel  to  it,  the  supplementary  chord  PB,  and  through  D 
draw  MT  parallel  to  PB. 

*  students  who  have  not  read  the  CaJcnluB  will  omit  this  paragrj^h. 


SPECIAL  PKOPERTIES  OP  THE  CONIC  SECTIONS. 


191 


To  draw  a  tangent  parallel  to  a  given 

line,  or  what  is  the  same  thing,  making 

a  given  angle  with  the  axis  of  x,  draw 

a  chord  P  B  parallel  to  the  given  line 

EF,  or  making  the  given  angle  PBX, 

draw    the    supplementary  chord    PC, 

and  the  diameter  D  D'  parallel  to  PC. 

Through  D  draw  MT  parallel  to  PB 

and  it  is  the  tangent  sought. 

Fig.  207. 

SjLS,  Cor.  2. — If  one  of  two  supplementary  chords  to  the  transverse 
axis  is  parallel  to  a  diameter,  the  other  chord  is  parallel  to  the  conjugate 
diameter,  for  the  latter  diameter  is  parallel  to  a  tangent  at  the  vertex  of  the 
former  {3 OS).  Hence,  also,  if  one  diameter  is  conjugate  to  another, 
reciprocally,  the  latter  is  conjugate  to  the  former. 

Ex.  1.  In  an  ellipse  whose  axes  are  8  and  6,  one  supplementary 
chord  to  the  transverse  axis  makes  an  angle  with  that  axis  whose 
tangent  is  2  ;  what  angle  does  the  other  make  ? 

Solutions. 

the   formula 

A- 


-Arithmetically.  Using 
aa'  =  —  -^,  a  =  2, 


J5  =  3,  and  J.  =^  4 ;  whence  a'  = 

9 
—  — ;     or  the    angle  is    164°  18' 

nearly.        Geometrically.    Construct  0 

the  ellipse  with  C  B  =  8  and  D  E 

=  6.     Make  PCB  =  tan-'  2  and 

draw  P  B.     The  angle  P  B I  is  the 

cue  reqmred,  whose  tangent  is  found 

9 
by  measurement  to  be  about . 


Fig.  208. 


Ex.  2.  The  same  as  above,  the  tangent  of  the  angle  being  —  5,  and 
the  axes  10  and  6, 

Ex.  3.  The  same  numbers  as  in  Ex.  1,  applied  to  the  hyperbola. 
What  are  the  co-ordinates  of  the  point  in  the  curve  from  which  the 
chords  are  drawn? 

_B2  9 

SuG. — ^For  the  solution  of  the  last  question  we  have  aa'  =  —    or  a  =  — ,  a' 

A^  32 

9 
being  2  ;  y  =z2{x  — 4)  and  y  =  —(a;  -f-^)  to  find  x  and  y,  which  are  nearly  5.3 

and  2.6. 

Ex.  4.  In  an  ellipse  whose  axes  are  8  and  6  find  the  angle  included 
by  the  supplementary  chords  to  the  transverse  axis,  from  the  point 
X  =1. 


192 


PKOPERTIES   OF  PLANE   LOCI. 


3  IS,  JPvoh, —  To  draiv,  geometrically,  a  pair  of  supplementary 
chords  to  the  transverse  axis  of  a  given  ellipse  so  that  the  chords  shall 
include  a  given  angle. 

Solution.  — Upon    the  -  ^ 

transverse    axis    describe  p^ 

a  segment  of  a  circle 
B  P  M  A  which  shall  con- 
tain the  given  angle — in 
this  case  ABC.  From 
the  intersection  of  this 
circumference  with  the 
I  llipse,  as  from  P  or 
P',  draw  supplementary 
chords.      The  pupil  may  Fig.  209. 

give  the  reason,  and  also  show  how  the  consDruction  will  indicate  the  impossibility 
in  case  the  angle  given  is  larger  or  smaller  than  can  be  included  by  chords  in  the 
given  ellipse. 

317'  GsNEKAii  Scholium. — ^From  the  preceding  articles  in  this  section 
it  is  evident  : 

1st.  That  to  draw  a  diameter  conjugate  to  a  given  one,  we  may  draw  a 
tangent  through  the  extremity  of  the  given  diameter  and  then  draw  a  di- 
ameter parallel  to  this  tangent ;  or  we  may  draw  one  of  two  supplementary 
chords  parallel  to  the  given  diameter,  and  dravv'ing  the  other  supplementaiy 
chord,  draw  the  second  diameter  parallel  to  the  last  chord. 

2nd.  To  draw  a  pair  of  cc  njugate  diameters  which  shall  include  a  given 
angle,  draw  a  pair  of  supplementary  chords  which  shall  include  the  angle, 
and  parallel  to  these  draw  a  pair  of  diameters. 

3rd.  If  a  be  the  tangent  of  the  angle  -svhich  one  diameter  makes  with  the 
transverse  axis  and  a    the  tangent  of   the  angle  which   the  other    makes, 


aa  = 


5^ 


in  the  ellipse,  and  aa 


T2 


in  the  hyperbola.    Letting  tan-'  a 


sin  oc                   sin  a' 
and  tan""'  a'  be  respectively  a  and  oc' ,  and  w^e  have  a  =  — —  and  a'  = . 

cos  a  cos  a  > 

Substituting  these  values,  there  results  in  the  case  of  the  ellipse  ^-sin  a  sin  a' 
4-  -S-  cos  oi  cos  oc'  =r:  0,  and  of  the  hyperbola  A'^  sin  a  sin  a  —  B-  cos  oc  cos  oc' 
=  0,  formulcE  which  are  sometimes  referred  to  as  the  squations  of  condition 
of  conjugate  diameters. 


dth.  From,  the  relation  aa 


B^ 

—  we  may  also  solve  the  2nd    above. 


Thus,  if  (j  be  the  given  included  angle,  (S  =  a  —  oc,  and  tan  /?  = 


B^ 

A-^a 


1  -j-  aa 


supposed  known.. 


whence  a'  can  be  determined,  as  all  the  other  quantities  are 


SPECIAL   PROPERTIES    OF   THE   CONIC    SECTIONS. 


193 


318.  JProb* — To  investigate  the  relation  between  conjugate  diameters 
and  the  axes. 

Dem.  — The  equation  of  the  ellipse  and  hyperbola  referred  to  the  conjugate 
diameters  2^2  and  2^2  is  A^^^y^  ±  Bz^x^  =  ±  A-y^B^^  {127,  jEx's.  10  and  11)  ;  the 
4-  sign  applying  to  the  ellipse  and  the  —  sign  to  the  hyperbola.  Transforming  this 
equation  so  that  the  reference  shall  be  to  the  axes,  by  means  of  the  formulce 


2/2 

we  have 

Az^y^cos'^a 
zb  B-i'^y-  cos-  a 


y  cos  a.  —  a;  sm  a 


sm  {a 


a) 


X2  — 


X  sm  a 


y  cos  a 


sin  (a 


a) 


{127,  ^cm), 


±  A2^B2^Bin^{a'  —  a). 


-  2^2 ^^y  cos  asina  -\-  Az^x^ sin^  a 
2B.2'xy  cos  a'sin  a'  ±  B^^x-  sin2  a' 
Comparing  this  with  A'^y'^  ±  B^x:^  ^  ±  A^B%  we  have 

(1)  A2  2  cos2  a  ±  -B22  cos'^a'  =  J.2, 

(2)  Az'-'sin^a  ±  B^^sin^a'  =  ±  ^2, 

(3)  Az'^  cos  a  sin  a  ±  5^2  cos  a'  sin  a  =  0,  and 

(4)  Az^Bz'^sm'^  {a   —  a)  =  A^B^. 

1.  Adding  (1)  and  (2)  and  remembering  that  sin2  -j-  cos2  =  1,  we  have 
A^^  =h  B^^  =^  A^  ±  B%  that  is 

(a)  In  the  ellipse  the  sum  of  the  squares  of  conjugate  diameters  is  constant,   and 

equal  to  the  sum  of  the  squares  on  the  axes.    In  the  hyperbola  the  difference  of  the 

squares  is  constant  and  equal  to  the  difference  of  the  squares  on  the  axes. 

cos  a  sin  a         _  .      ,     ,  .         ,  ^ 

-h  1  ;   but  as  ^2  and  B^  ^^^  conju- 


2.  Making  A2'^=B2'^  (3)  gives 


cos  a  sm  a 


gate   tan  a  tan  a'  = 


smasma  B^      ^^  ■,,.-,  .        ,,  ,  „   ., 

;  =  -t-   -— .      Multiplymg  the  members  of  these 

cos  a  cos  a  A^ 


equations  and  rejecting  common  factors  we  have 


sin2  ^ 


J52          sm  a 
—-,  or  

A'^         cos  a' 


B 

A' 


the  —  sign  characterizing  the  ellipse  since  a'  is  obtuse  in  the  ellipse,  and  the  -f- 
sign  characterizing  the  hyperbola,  as  in  it  a  and  a'  are 


acute.     Hence 


B 


cos  a 


sm  a 


indicates  the  position 


cos  a 


of  equal  conjugate  diameters  in  the  ellipse,  and 

=  —. in  the  hyperbola.     From  Fiq.  210  we  see  that 

sm  a: 

A  D  and  A  F  meet  this  condition  in  the  ellipse  ;  for 

AC  A  ,    .  .  .  HB 


AK  = 

B 


sin  a 


cos  KAC       — cos  a'' 
Hence 


and  AH 


sin  H  A  B 


Fiq.  210. 


(b)  In  the  ellipse  the  conjugate  diameters  which  fall  upon  the  diagonals  of  the  rect-i 

angle  on  the  axes  are  equal,  and  a  and  a'  are  supplementary. 

A  B 

3.  In  the  hyperbola  in  general,  the  condition ;  =  -: —  is  met  only  when  the 

cos  a:        sma 

two  diameters,  as  G  F  and  DE  Fig.  212,  coincide  and  fall  on  the  asymptote. 
Hence,  in  the  hyperbola,  asymptotes  are  the  analogues  of  the  equal  conjugate 
diameters  of  the  ellipse.  But  from  -^22  —  B^^  =  A'^  —  B^,  we  observe  that  if 
A  =:s  B,  A2  =  Bz,  independently  of  a  and  a.     Henc6 


194 


PROPERTIES   OF   PLANE   LOCI. 


(c)  In  the  hyperbola,  in  general,  there  are  no  equal  {finite)  conjugate  diameters  ;  hut, 
in  the  equilateral  hyperbola,  any  pair  of  conjugate  diameters  are  equal  each  to  each. 
The  conjugate  diameters  of  the  equilateral  hyperbola  find  their  analogues  in  the 
diameters  of  a  circle. 

4.  Making  A2  =  B^  in  A^^-  -}-  B^^  =  A^  -{-  B%  we  find  that  J^   

(d)  The  length  of  one  of  the  equal  conjugate  diameters  of  an  ellipse  is  \/2  v/A^-j-B--^. 
.-.    The   semi-conjugate   diameter:     the    semi-diagonal  on  the  axes  ::  1  :  V^. 

5.  Extracting  the  square  root  of  both  members 
of  (4),  we  have  A^^Bz  sin  (a'  —  a)  =  AB  ;  .which 
signifies  that 

(e)  The  parallelogram  formed  by  tangents  drawn 
through  the  vertices  of  any  pair  of  conjugate  dia- 
meters is  constant,  and  equal  to  the  rectangle  on  the 
axes.  This  will  be  more  apparent  from  Fig's.  211, 
212.  D  A  F  =  (a'  ~  a),  and  D  A  =  ^5^, ;  whence 
AO  =  -S2  sill  (<^' —  ^)-  Hence  J.2-B2  sin  {a'  — a) 
=  area  AFHD  =  iLKIH  =  AB  =  i  the 
rectangle  on  the  axes.     .  • .    L  K I  H  =  the  rectangle  on  the  axes, 

Ex.  1.  Write  the  equation  of  an  el- 
lipse referred  to  a  pair  of  conjugate 
diameters  whose  lengths  are  8  and  6, 
and  the  included  angle  tan~^  ( —  2). 
Having  written  the  equation  con- 
struct it  as  in  Chapter  I.,  Sec.  II. 

Ex.  2.  Write  the  equation  of  an 
hyperbola  referred  to  conjugate  dia-  ^ig*  212. 

meters  whose  lengths  are  12  and  8,  and  whose  included  angle  is 
tan~^2.     Construct  as  in  the  last  example. 

Ex.  3.  In  an  ellipse  whose  axes  are  8  and  6  what  is  the  length  of  a 
diameter  which  makes  an  angle  of  45°  with  the  axis  oi  x?  What  is 
the  length  of  its  conjugate  ? 

Stjg's. — ^From  the  relation  aa'  = j  >  we  learn  thart  the  conjugate  diameter  makes 

with  the  axis  of  x  an  angle  of  150°  39'  nearly.     Hence  A2B2  sin  {a'  —  a)  =  AB, 

becomes  ^12^2  sinl50°  39'  =  12,  or  AoB.  =  ~J^.    AlsoA^^  4-  -822  =  ^2  +  B^ 

.49014  2     r      z  I 

gives  A2^-i-B2~  =  2^5.     These  two  equations  will  give  the  values  of  A2  and  Bz- 

Ex.  4.  In  an  ellipse  whose  axes  are  8  and  6,  what  are  the  sides  of 
the  circumscribed  parallelogram  whose  sides  are  parallel  to  the  equal 
conjugate  diameters  ?     What  is  the  altitude  of  this  parallelogram  ? 

Altitude,  6.79  nearly. 


SPECIAL  PEOPERTIES  OF  THE  CONIC  SECTIONS. 


195 


(€)    PROPEETIES  OF  ORDINATES. 

310.  JPvop, — The  squares  of  ordinates  to  the  transverse  axis  of  an 
ellipse  are  to  each  other  as  the  rectangles  of  the  segments  into  ivhich  they 
respectively  divide  the  axis. 

Dem.— Let  PD  =y,  FD'  =  ?/'.  AD  = 

jc,  A  D'  =  X',  then  A-y^'  +  B^x^=  A'^B'^  and 
A-y'^   -f-   B-^x''^   =   A^B^ ;     whence    y^  = 

B^  B-^ 

-—  {A^  —  x2)  and  y'^  =  -r;( A^  —  x'^).     Divid- 

A^  A~ 

ing  and  rejecting  the    common   factor  we 

have  ^~  = =  ^—-^, — -^^ -',  or 

yi      ^2  _  a;2  ^A-\-  x){A  —  x) 

y2  : 2/'2  : :  (^  +  x){A  —x):{A-\-  x'){A  —  x') 

or    ::    CD     X     DB     :    CD'    X     D'B. 

^•^•^-  Fig.  213. 

320,  Cor.  1. — The  square  of  any  ordinate  to  the  transverse  axis  of 
an  ellipse  is  to  the  rectangle  of  the  segments  into  which  it  divides  that  axis, 
as  the  square  of  the  conjugate  axis  is  to  the  square  of  the  transverse. 

Dem.— In  the  above  proportion  if  y'  ==  GA=B,  {A  -\-  x'){A  —  x')  =  A-,  and  we 
have  2/2  :  -B2  : :  C D  X  D B  :  ^2.     ...  y2  :  CD  X  DB  :  :  4:B-i  :  4:A^     q.  e.  d. 

321,  CoE.  2. — The  latus  rectuyn  is  a  third  proportional  to  the  trans- 
verse and  conjugate  axes. 

Dem.— In  the  last  proportion  let  y  become  the  focal  ordinate  P"F,  which  call  p, 
and  CD  X  DB  becomes  CF  X  FB  =  {A-\-  c){A  —  c),  c  being  AF.  Now 
{A  +  c){A  —  c)  =  ^2  —  c2  =  B^  hence  p^  iB'^  ::  B'^  :  A%  or  2A  :2B  ::2B  :  2p. 

Q.  E.  D. 

322,  ScH.— The  prop- 
erties demonstrated  in 
this  proposition,  and  in 
the  1st  and  2nd  corolla- 
ries, are  equally  true  for 
the  hyperbola,  and  can 
be  proved  in  the  same 
way.  In  the  case  of  the 
hyperbola,  however,  the 
statement  should  be, 
The  rectangles  of  the  dis- 
tances frota  the  feet  of 
the  ordinates  to  the  ver- 
tices, instead  of  "the  ^^^'  '^^'^' 
rectangles  of   the  segments,   etc.,"  as,  in  this  case   the   ordinates  do  not 


196 


PBOPEKTIES   OF   PLANE  LOCI. 


divide  the  axis,  but  fall  upon  its  prolongation ;   so  that,  in  Fig.  214,  we 
have  y2  :  y'2  ; ;  qq  x  BD  :  CD'  X  BD'. 

323,  CoE.  3. — In  the  case  of  the  circle  Coe.  1st  shows  that  the  square 
of  the  ordinate  equals  the  rectangle  of  the  segments  into  which  it  divides 
the  diameter — a  well  known  property. 

324:.  Coe.  4. — This  proposition  and  Coe,  1st  may  be  asserted  of 
ordinates  to  the  conjugate  axis.  [Let  the  student  give  the  proof  and  a 
figure  to  illustrate  it.] 

32S,  Coe.  5. — This  pj^oposition  and  Coe.  1st  may  also  be  asserted  of 
ordinates  to  any  diameter  of  an  ellipse  or 
an  hyperbola. 

Dem. — The  corollary  can  be  proved  in  the 
same  way  as  the  proposition,  by  using  the 
equation  of  the  curves  referred  to  conjugate 
diameters  {127 f  Ex's.  10  and  11),  since  these 
equations  are  of  the  same  form  as  those  used 
above.  In  the  annexed  figures,  therefore, 
PD'  :  P  D  '  ::  CD  X  DB  :  CD'  X 
D'B.    Also  PD'  :  CD  X  DB  ::^2  :^'2. 


Fig.  215. 


326.  Coe.  Q^.—From 
the  last  relation,  it  fol- 
lows that  chords  paral- 
lel to  any  diameter  are 
bisected  by  its  conjugate, 
i  e.PD  =  DH,P'D' 
=  D'H',  etc.  ;  and 
hence  that  these  curves 
are  symmetrical  with  re- 
spect to  any  diameter. 


327*  ScH. — These  principles,  together 
with  others  already  known,  enable  us  to 
find  the  centre,  axes,  and  foci  of  the  curves, 
geometrically,  when  the  perimeters  alone 
are  given.  Thus,  in  the  case  of  the  ellipse, 
let  the  curve  NHIM  be  given,  to  find  the 
centre,  axes,  and  foci.  Draw  any  two  par- 
allel chords  as  DE  and  BO,  bisect  them 
at  K  and  L,  and  draw  FG  ;  it  Avill  be  a 
diameter  by  Coe.  6.  Bisect  this  diameter 
and  A  will  l)e  the  centre.     From  A  with 


Fig.  216. 


Fig.  217. 


SPECIAL  PROPERTIES  OF  THE  CONIC   SECTIONS. 


197 


a  radius  sufficiently 
long  to  cut  the  curve, 
construct  the  circle 
HIMN,  join  two  of  the 
intersections,  as  I  and 
H,  and  perpendicular 
to  this  chord  pass  a  line 
through  the  centre  ;  it 
will  be  the  axis.  [The 
student  can  readily  fin- 
ish the  problem.] 

The  construction  is 
the  same  for  the  hy- 
perbola except  in  find- 
ing the  conjugate  axis 
when  the  conjugate 
hyperbola  is  not  given.  Fig.  218. 

For  this  purpose  use  the  proposition  in  Cok.  1.     In  the  figure,  take 

SR  X  VR  :  HR^  :  :  AV'  :  AO'^  ;   whence  AO  can  be  constructed. 


S2S,  IProp. — In  different  ellipses  upon  the  same  transverse  axis,  the 
corresponding  ordinates  to  the  transverse  axis  are  to  each  other  as  the  con- 
jugate axes  of  the  respective  curves. 


Dem.— We  have  PG  :  CG  X 


GB  : 

:AD^ 

:AB' 

alsoP'G'  : 

CG  XGB: 

:  AD 

':  AB'  and 

P  G^ 

:  CG 

XGB::  AD'": 

ABl 

.•  .  PG  :  PG  :  P"G  :  : 

AD  : 

AD': 

AD    , 

etc.    Q.  E.  D. 

329,  Co^.—Amj  ordi- 
nate to  the  transverse  axis  of 
an  ellipse  is  to  the  correspond- 
ing ordinate  of  the  circum- 
scribed circle  as  the  conjugate 
axis  of  the  ellipse  is  to  the 
transverse. 


[ 

y" 

Sf" 

. 

Z-^^^^Z— 

dJ[^ 

p>\ 

/^ 

-^^^^1_— 

D'       / 

P^\^\ 

(^ 

D     / 

P^^^^\\\ 

c 

// 

1 

A 

V 

G                   J 

-^x^ 

"■^^^-—^    H 

1 

^^^-^^^y/ 

"^ 

^^ 

B 


Fig.  219. 


Dem. — Let  CD"'B  be  the  circumscribed  circle,  then  as  it  may  be  considered  as 
an    ellipse   with    equal    axes,   we    have  PG    :  P"'G    ::  AD    :  AD"'(=  AB% 

Q.   E.    D. 


198 


PROPERTIES  OP  PLANE  LOCI. 


330,  ScH. — An  instrument  called  a 
Trammel  is  constructed  upon  the  prin- 
ciple enunciated  in  this  corollary.  It 
consists  of  two  grooved  bars  X '  X ,  Y  Y ', 
fastened  together  at  right  angles,  and 
an  adjustable  arm  PH.  H  and  I  are 
pins  which  can  be  fastened  anywhere 
on  PH,  and  have  heads  on  the  under 
side  which  run  in  the  grooves  of  the 
bars.  Any  point  in  the  movable  bar, 
as  P,  traces  an  ellipse  as  H  and  I  slide 
back  and  forth  in  the  grooves.  To 
prove  that  P  is  a  point  in  an  ellipse  of 
which  PH  is  the  semi-transverse  axis  and  PI  the  semi-conjugate,  draw 
AP"'  and  PH  parallel  to  it,  Fig.  219.  Produce  PG  till  it  meets  HE  drawn 
paraUel  to  AB,  in  E.  Then  AP'"  =  ^  =  PH.  Again,  P"  G  :  PG  : : 
P"'A  :  PI,  or  ordinate  of  circle  :  ordinate  of  ellipse  :  :  A  :  PI.  And  as  this 
is  true  for  all  positions,  PI  being  made  =  B,  and  PH  =  ^,  P  is  always 
in  the  curve. 

We  may  also  demonstrate  directly  that  the  locus  of  P  is  an  ellipse.     Prom 
Fig.  220,  using  the  common  notation  PI:PH  ::PD:PE,  gives  B  :  A  :'. 

y  :  >/ J.2  —  x^,  or,  squaring,  B^  :  A^  : :  y^  :  A-2  —  x^ ;  whence  A^y^  +  B^x^  =^ 
A^B\ 


331,  I^TOp, — In  different  ellipses  on  the  same  conjugate  axis,  cor- 
responding ordinates  to  this  axis  are  to  each  other  as  the  transverse  axes 
of  the  respective  curves. 


Dem 
GE  .:  AB    :  AD 
DG  ~ 


We  have  PG  :DG  X 
and  p'Q- ; 


J/2 


X   G  E  :  :  A  B' 
.-.    PG  :    PG  ::  AB 

Q.   E.   D. 


AD^ 
:  AB' 


332,  Cor. — Any  ordi- 
nate to  an  ellipse  is  to  the  cor- 
resjionding  ordinate  of  the  in- 
scribed circle,  as  the  transverse 
axis  of  the  ellipse  is  to  the  con- 
jugate. [The  student  may 
make  the  deduction  from 
the  proposition.] 


SPECIAL  PEOPEETIES   OF   THE   CONIC    SECTIONS.  199 

333,  I*rop* — The  squares  of  ordinates  to  any  diameter  of  a  para-r 
hola  are  to  each  other  as  their  corresponding  abscissas. 

Dem. — Referred  to  any  diameter,  as  AX  or  AjXi, 

the  equation  of  the  parabola  is  y^  =  2px  {127 f  ^x,.  12). 
Whence,  letting  y  and  y'  represent  any  two  ordinates, 
as  PD  and  P'D,  or  PiDi  and  P^'D]',  and  .-r  and 
;c'  the  corresponding  abscissas  we  have  y~  =  2px  and 

2/ '2  ■=  2px'.     Dividing,  — -  =  -^.     q.  e.  d, 

334:,  CoK. — All  chords  drawn  parallel  to  a 
tangent  at  the  extremity  of  a  diameter  of  a  par-  ^ig  222 

abola  are  bisected  by  that  diameter. 

335.  ScH. — Having  the  curve  to  find  the  axis  and  focus  of  a  parabola, 
we  draw  any  pair  of  parallel  chords,  and  bisect  tliem  by  a  right  line.  This 
line  is  a  diameter.  Draw  two  other  parallel  chords  perpendicular  to  the 
diameter  thus  found,  bisect  these  chords  by  a  right  line,  and  it  will  be  the 
axis.     Find  the  focus  by  {164,  or  284). 


(d)   ECCENTEIC   ANGLE. 

336,  Def. — The  JEccentric  Angle  in  an  ellipse  is  the  angle 
formed  with  the  axis  of  abscissas  by  a  line  drawn  from  the  centre  to 
a  point  in  the  circumference  of  the  circumscribed  circle  where  a  pro- 
duced ordinate  meets  it,  that  is  P"'A  B  Fig.  219. 

337.  I*rop. — The  abscissa  of  any  p>oint  in  the  ellipse  equals  the 
semi-transverse  axis  into  the  cosine  of  the  eccentric  angle,  and  the  corres- 
ponding ordinate  equals  the  semi-conjugate  axis  into  the  sine  of  the  same 
angle.     That  is,  letting  q)  represent  the  eccentric  angle, 

X  =  A  cos  qp,   and  y  z=  B  sin  q). 

Dem.— In  Fig.  219,  AG  =  a:  =  P"'A  cos  P  " A  B  —  Acos  q).  Also  PG  =  y 
=  P I  sin  P I G  =  5  sin  9>. 

ScH. — The  introduction  of  this  angle  is  a  recent  device  to  facilitate  the 
deduction  of  certain  properties  of  the  ellipse.  It  enables  us  to  transform 
an  equation  in  terms  of  rectangular  co-ordinates  {x,  y)  into  one  containing 
but  one  variable,^,  which  is  sometimes  of  much  advantage.  We  will  give 
a  few  specimens  of  its  use. 

33 S.  I^vop, — The  equation  of  a  tangent  to  the  ellipse  in  terms  of 
the  eccentric  angle  is 

A  sin  cp  ■  y  -\-  B  cos  cp  ■  x  =  AB. 

Dem. — The  equation  of  a  tangent  to  an  ellipse  is  A^y'y  ■\-  B-x'x  =  A-B\  As 
(x',  y')  is  a  point  in  the  ellipse,  we  have  x'  ==  A  cos  cp,  and  y'  =  .Bsin  q).  Substi- 
tuting these  values  and  dividing  by  AB,  we  have  A  sin  cp-  y  -\-  B  cos  <p  '  x  ^^  AB, 

Q     E.   D. 


200  PROPERTIES  or  PLANE  LOCI. 

330,  I^vop, — The  eccentric  angles  of  the  vertices  of  conjugate  diam^ 
eters  differ  by  90°. 

Dem. — Let  D  AB  =  ^,  and  D' AB  =  ^',  be  the  ec- 
centric angles  of  the  vertices  of  the  conjugate  diameters 
PC  and  P'C.  Letting  (x,  y)  be  P,  and  (ccj,  y^)  be 
P',  the  equations  of    AP  and  AP'   are,   respectively 

?y  ?/ 

y  =  aXy  or  a  =  -,  and  y^  =  axi,  or  a   =  — .     Whence 

B^       yvi       B  sin  (p  X  B  sin  cp'       ,  ,         ,  -r^       ^.^o 

aa'  = -  =  ^^  = -^--^— -:; —,  or  tan  cpt&ncp  Fig.  223. 

A'^       xxi        AcoscpX  A  cos  cp' 

=  —  1.     Hence  A  D  and  A  D '  are  perpendicular  to  each  other,  and  (p'  =  q)  -{^ 

90°.      Q.  E.  D. 

34:0,  ScH. — This  proposition  affords  a  ready  method  of  drawing  a  diam- 
eter conjugate  to  a  given  diameter.  Thus  let  PC,  Fig.  223,  be  the  given 
diameter.  Circumscribe  the  circle,  produce  the  ordinate  PE  to  D,  draw 
DA,  and  D'A  perpendicular  to  it.  From  D'  let  fall  the  perpendicular 
D'E',  and  P'is  the  vertex  of  the  conjugate  diameter  required. 

34:1.  JPvop, — The  rectangle  of  the  radii  vector es  drawn  to  the  ex- 
tremity of  any  diameter  equals  the  square  of  the  semi-conjugate  diameter. 

Dem. — Let  F'P  =  r',  and  PF  =  r,  Fig.  223,  and  the  other  notation  remain 
as     before.        Then     from    F'PE    we    have    r'    =     \/y~   -f~    {Ae   -{-   xy-     == 

V{A2  —  x2)(l  —  e^)  4-  ^2e2  ^_  2Aex  -f  x^  =  V A^  +  2Aex  -f  e^^c^  =  A -\- ex.  In 
like  manner  from  PE F,  r  =  J.  —  ex.  Whence  rr'  =  A^  —  e-x"^.  Again  P'A  = 
j/i  2  -{-  ccjS  _  J52  sin2  (p'  -f  A^  cos2  qj'  =  (J.2  —  A^e^)  sins  cp'  -f-  J.2  cos  5^'  ==  A^  — 
A^e'^  sin^  (p' .  But  9>' =  90°  -j-  (p  ;  whence  sincp'  =  cos  cp,  and  P' A'^=  A-  — 
e2  •  A-cos'2  (p  ^=  A^  —  e-x^.     .' .  rr'  =  P'A^.    q.  e.  d. 

342*  JPvop,—The  sum  of  the  squares  of  any  pair  of  conjugate 
diameters  is  constant  and  equal  to  the  sum  of  the  squares  of  the  axes. 


Dem. — In  Fig.  223  we  have   P' A'  =^  x^-  -{-  y^^  =  B-sin^cp'  -f-  ^2cos2^'  ;    or 
since  (p'  =90° -|-^  sin  q)'  =  cos  cp,  and  cos  cp'^=  —  sin  cp, 

■ — -, 2 

P'A    =  A^  sin2  q)  -^  B-  cos^  qj  ;  and  in  like  manner, 
P  A'  =  ^-  cos-  q)  4-  B-  sin2  q). 


Adding    P  A'  +  P'  A"  =  A"-  -{-  B^-.  Multiplying  by  4, 


4PA'  + 4P'A'=:4^2_|_4^e.     Q.  E.  D. 

ScH. — This  proposition  has  been  demonstrated  before  {318 ^  a),  but  is 
inserted  here  as  its  demonstration  affords  an  example  of  the  utility  of  the 
eccentric  angle. 

Ex.  1.  What  is  tlie  eccentric  angle  of  the  extremity  of  the  trans- 


SPECIAL  PBOPERTIES  OF  THE  CONIC  SECTIONS. 


201 


verse  axis  ?     What  of  the  extremity  of  the  latus  rectum  ?     What  of 
the  extremity  of  the  conjugate  axis  ? 

Ans.,  (p  =  0°,  cp  =  cos~^  e  =  sin~^  —  cp  =  90®. 

Ex.  2.  In  an  elHpse  whose  axes  are  8  and  6,  what  is  the  eccentric 
angle  at  :r  =  1  ?  What  are  the  co-ordinates  of  the  point  of  which 
the  eccentric  angle  is  60°  ?     45°  ?     30°  ? 

Ex.  3.  In  an  ellipse  whose  axes  are  12  and  8  what  is  the  length  of 
the  diameter  from  the  point  whose  eccentric  angle  is  60°  ? 

SuG.— Calling  the  semi-diameter  A^  we  have  A2- =  A'^  cos^  g) -\-  B'^sin'^q)^ 
36  X  {ky  -+-  16  X  (iv'3)2  =  21,  and  A^  =  v/21. 


343,  JProp, — The  intercepts  of  a  secant  between  the  hyperbola  and 
its  asymptotes  are  equal. 

Dem. — Let  DD'  be  any  secant,  and  P 
and  P',  the  points  in  which  it  cuts  the 
curve,  be  designated  respectively  as  {x,  y') 
and  (x",  y").  Since  DD'  is  a  line  pass- 
ing through  the  two  points  (x,  y'),  and 
{x",  y"),  we  have  for  its  equation  y  —  y'  = 


y 


y 


-{x — x).     And   since    (.r',  y')   and 
X   —  cc" ' 

{X",  y")  are   points  in   the   curve  x'y'   = 

x"y"  =  m.      If  in  the  equation  of   D  D ' 

we  make  y  =  0,  x  =  AD,  and  x  —  x'  = 

CD.      Hence  we  have    CD  =  x  —  x'  = 


X  y  —  y  X 


Fig.  224. 

_  x"y"  —  y'x" 

y"  —  y' 


=  X"   = 


y"  —  y' 

C'P'.     Now  as   PCD  and  P'C'D'  are  equiangular  and  have  CD  =  C'P', 
the  triangles  are  equal,  and  PD  =  P'  D'.     Q.  e.  d. 

ScH. — This  proposition  afibrds  an  ele- 
gant and  convenient  method  of  construct- 
ing the  hyperbola.  If  the  axes  are  given, 
put  them  in  position  and  draw  the 
asymptotes,  which  are  the  diagonals  of 
the  rectangle  on  the  axes.  Then,  through 
the  extremities  of  the  transverse  axis, 
draw  a  convenient  number  of  radiant 
lines,  as  aa' ,  'hh' ,  cc\  dd\  and  make  the 
intercepts  \a' ,  2b',  3c',  4c?'  respectively 
equal  to  Ba,  Bb,  Be,  Be?.  Then  are  1,  2, 
3,  4,  points  in  the  curve. 

If  the  asymptotes   are   given,  or  the 


Fia.  225. 


202 


PROPERTIES   OF   PLANE   LOCI. 


angle  included,  and  any  point  in  the  curve  as  P,  the  asymptotes  can  be 
drawn  ;  and  then  radiant  hues  through  P  will  be  secants  whose  intercepts 
will  make  known  points  in  the  curve. 

PARAMETER  TO  ANT  DIAMETER. 

o44.  A  Parameter  to  any  iPiameter  of  an  Ellipse  or 
Hyperbola,  is  a  third  proportional  to  that  diameter  and  its  conjugate. 
In  the  Parabola  it  is  a  third  proportional  to  any  abscissa  and  its  cor- 
responding ordinate. 

34:S,  J^rop, — The  distance  from  any  point  in  a  Parabola  to  the 
focus  is  one  fourth  the  parameter  to  the  diameter  from  that  point. 

Dem.— Let  A2  F  =/ ;  then  is  y2^  =  ¥^z-     From  Ex.  12,  page  88,  we  have 2/2 2 


2p 
=    ■   ,    X2  j  and  also  2n  sin  a'  —  2r)  cos  a' 


0. 


From  the  latter,  n^  sin^o:' 
p2sin"a';    whence   sin^a' 


p2  cos'-'a'  =  p2  — 
p2 


n2-f  p2  • 

2p  2(7i2  4-  p2)  2(2pm  -f-  p2) 


Hence 


sin-'a'  p  p 

4(to  -}-  ip),  since  n^  =  2pm.  But  m  -f-  sP  = 
TF  =  FA2  =  /.  Therefore  2/2^=  ¥^2, 
or  iCg  :  2/2  •  •  2/2  '■  ¥ '}  and  4/  is  the  parameter 
to  the  diameter  A. 2^2     Q.  e.  d. 


Fig.  226. 


34:0,  CoR.  1.— The  parameter  to  any  diameter  of  a  Parabola  is  four 
times  the  distance  from  the  vertex  of  that  diameter  to  the  directrix. 

34:7*  CoE.  2. — The  double  ordinate  to  any  diameter  of  a  Parabola, 
which  {ordinate)  passes  through  the  focus,  is  the  parameter  to  that  dia- 
meter. 

Dem. — Let  AgH  =X2,  and  LH  =  2/2 1  whence  2/2^=  ^f^i-  Now  A^H  =; 
Xi=  TF  =  A2F  =/.  Wherefore  ?/2 2  =  4/2  ;  and  2/2  =  2/.  But  IL  = 
2LH  =  2y.2.  =^  4f.     . • .  I  L  is  the  parameter  to  A2a;2' 


34S,  JProp, — Any  chord  ichich  passes  through  the  focus  of  an 
Ellipse  is  a  third  proportional  to  the  transverse  axis  and  a  diameter  par- 
allel to  the  chord. 

Dem.— Let  PF=:r,  PFB  =  a:,  and  P'F  =  r'.  Then 

P  _„,-,  ...  V 


-,  and  r'  =  _    , 
1  —  e  cos  a  1-j-ecosa: 


{107)',  whence  r 


4-r'=  PP'  =: 
86,  5i2    = 


2p 


1  —  e-  coti'^a 


But  from    JEx.  10,    page 

AHl  —  e2) 


A'^  siii'^a  -f-  B^  cos- a 


e-  cos- a 


P' 
Fig.  227. 


Ap 


1  —  e^  cos^a 
2 


SPECIAL  PROPERTIES  OF  THE  CONIC   SECTIONS. 


;  by  substituting  A^(l  —  e^)  for  B^  and  reducing.    Therefore 


203 
pp_ 


,  or  2  J. :  BjCi  :  :  BiCj  :  PP'.     q.  e.  d. 

349.  ScH. — Tlie  statement  in  [347)  is  not  true  in  case  of  the  ellipse,  as 
will  appear  from  this  proposition. 


CHORD   OF   CURVATURE. 

[Note.— The  following  proposition  is  designed  to  be  read  by  those  who  have  taken  the 
Dlfterential  Calculus,  and  have  studied  Section  VI,  Chapter  IV,  or  have  some  knowledge  of  the 
subject  of  radius  of  curvature.] 

350.  A  Chord  of  Curvature  is  a  chord  of  the  Osculatory 
Circle,  drawn  from  the  point  of  contact. 

351.  I^rop, — In  the  parabola,  the  chord  of  curvature  which 
passes  through  the  focus  is  the  parameter  to  the  diameter  passing 
through  the  point  of  contact. 


Dem.  O  being  the  centre  of  the  os- 
culatory circle  at  3P,  in  the  parabola 
whose  focus  is  "F,  IE*^wC  is  the  chord  of 
curvature  passing  through  the  focus,  and 
is  the  parameter  to  IP3D,  the  diameter 
through  IE*.  For,  draw  IFXj  perpendic- 
ular to  the  tangent  through  P,  and  we 
have  from  the  similar  triangles  Jr^JbdiiMC 
and  1P:E*'Lj, 

■F:E<,  :  lE^IS^C  : :  IT  :  FI-.,  or. 


:!E*1^  = 


-p-p 


2N^ 


But  1*1^  =  — j »,  I[  being  the  normal, 


and  p  the     semi-latusrectum  of  the  curve  {211);  !FIj  =  ^!F*IEj  =  \]Sf  {164: 
or  284),  and  IT  =  A/^^  +  XJF 


*'  =  \/\n^-\-~N'^1 


4pi 


pi 


1  /2>2  4-  ^2  7^2 

g  N'Y  ^     /    -  —  {143,  Ex.  2),  remembering  that  p^  +  y^  =  W\     Substituting 


these  values,  we  have,  IPIMI  =  — s-  x  -pr- 


2i^3      ]^       2p       2i\r2 


p' 


2       W' 


= =  4Jr'Jb'   and  hence  is 

P 


the  parameter  to  IE*ID  {340). 


204  GENEEAL  SCHOLIUM. 

352,  CoK. — The  chord  l»s  intercepted  on  the  diameter  I»ID  is 
equal  to  the  chord  of  curvature  j^assing  through  the  focus,  since 
angle  CI*ID  =  TI^OSJ:. 

EKD    OF    PART    FIRST. 


GENERAL  SCHOLIUM. 

Book  Second,  treating  of  Loci  in  Space,  is  reserved  for  a  second  volume. 
The  present  volume  is  deemed  sufficient  for  the  use  of  all  students  in  our 
colleges,  except  such  as  pursue  mathematical  studies  as  a  specialty.  Yol- 
ume  n.  -^11  contain  Loci  in  Space,  and  a  more  extended  course  in  the 
Calculus. 


THE 

INFINITESIMAL  CALCULUS. 


INTRODUCTION. 

[>ToTE. — The  four  following  chapters  on  the  Diflferential  Calcixlus  are  to  be  read  iniinediately 
after  the  first  three  chapters  of  the  General  Geometry,  that  is,  the  first  92  pages  of  this  volume.] 

1,  Qtiatltity  is  the  amount  or  extent  of  that  which  may  be 
measured  ;  it  comprehends  number  and  magnitude.  (See  Akt.  4, 
General  Geometry,  and  the  two  SchoHums  under  it  on  pages  1 
and  2.) 

2,  NlilfYlbeT  is  quantity  conceived  as  made  up  of  parts,  and 
answers  to  the  question,  "How  many?"  (See  Art.  5,  Illustration, 
General  Geometry.) 

S*  Number  is  of  two  kinds,  DiscontiTlttOUS  and  CoTltin- 
uous. 

4,  Discofltiflttous  JVtcmber  is  number  conceived  as  made 
up  of  finite  parts ;  or  it  is  number  which  passes  from  one  state  of 
aggregation  to  another  by  the  successive  additions  of  finite  units, 
i,  e.,  units  of  appreciable  magnitude. 

S»  ContiflUOUS  JVttmber  is  number  which  is  conceived  as 
composed  of  infinitesimal  parts  ;  or  it  is  number  which  passes  from 
one  state  of  value  to  another  by  passing  through  all  intermediate 
values,  or  states. 

Ill's. — The  method  of  conceiving  number  with  which 
the  pupil  has  become  familiar  in  arithmetic  and  algebra, 
characterizes  discontinuous  number.  Thus  the  number 
1 3  is  conceived  as  produced  from  5  by  the  successive  ad- 
ditions of  finite  units,  either  integral  or  fractional.  In 
either  case  we  advance  by  successive  steps  oi  finite  length. 
If  we  say  5,  6,  7,  etc.,  tiU  we  reach  13,  we  pass  by  one  •^^^-  ^^ 

kind  of  steps;  and,  if  we  say  5.1,  5.2,  5.3,  etc.,  till  we  reach  13,  we  pass  by 
another  sort  of  steps  {tenths),  but  as  really  hy  finite  ones.  If,  however,  we  call  the 
hne  A  B,  Fig.  1,  x,  and  C  D,  x',  and  conceive  AB  to  slide  to  the  position  CD, 
increasing  in  length  as  it  moves  so  as  to  keep  its  extremities  in  the  lines  O  M  and 


B  D       '* 


B  D 

Fig.  3. 


2  INFINITESIMAL   Cx\LCULUS. 

O  N ,  it  will  pass  by  infinitesimal  elements  of  growth  from  the  value  x,  te  the  value 
x'  ;  or,  it  will  pass  from  one  value  to  the  other  by  passing  through  all  intermediate 
values,  and  thus  becomes  an  illustration  of  continuous  number. 

Again,  if  the  line  A  B,  Fig.  2,  be  considered  as  gen-       ^  

erated  by  a  point  moving  from  A  to   B,  and  we  call     AC  B 

the  portion  generated  when  the  point  has  reached  C,    •  ^^^'  2- 

X,  and  the  whole  line  x',  x  will  pass  to  x' ,  by  receiving"  infinitesimal  increments, 
or  by  passing  through  all  states  of  value  between  x  and  x'. 

A  surface  may  be  considered  as  generated  by  the  mo- 
tion of  a  Hue,  and  thus  afford  another  illustration  of 
continuous  number.  Thus  let  the  parallelogram  AF 
be  conceived  as  generated  by  the  right  line  A  B  moving 
from  AB  to  EF.  When  AB  has  reached  the  po- 
sition CD,  call  the  surface  traced,  namely  A  BCD, 
X,  and  the  entire  surface  A  B  E  F,  x'  ;  then  will  x  pass  to  x'  by  receiving  infinites- 
imal increments,  or  by  passing  through  all  intermediate  values. 

Finally,  as  volumes  may  be  conceived  as  generated  by  the  motion  of  planes,  all 
geometrical  magnitudes  -afford  illustrations  of  continuous  number. 

We  usually  conceive  of  time  as  discontinuous  number,  as  when  we  think  of  it  as 
made  up  of  hours,  days,  weeks,  etc.  But  it  is  easy  to  see  that  such  is  not  ttie 
way  in  which  time  actually  grows:  A  period  of  one  day  does  not  grow  to  be  a 
period  of  one  week  by  taking  on  a  whole  day  at  a  time,  or  a  whole  hour,  or  even 
a  whole  second.  It  grows  by  imperceptible  increments  (additions).  These  incon- 
ceivably small  parts  of  which  continuous  number  is  made  up  are  called  Infinites- 
imals. 

Motion  and  force  afford  other  illustrations  of  continuous  number.  In  fact,  the 
conception  which  regards  number  as  continuous,  vsdll  be  seen  to  be  less  artificial — 
more  true  to  nature — than  the  conception  of  it  as  discontinuous. 

6.  Jin  Infinite  Quantity  is  a  quantity  conceived  under  such 
a  form,  or  law,  as  to  be  necessarily  greater  than  any  assignable  quan- 
tity. 

7.  A.n  Infinitesimal  is  a  quantity  conceived  under  such  a 
form,  or  law,  as  to  be  necessarily  less  than  any  assignable  quantity. 

8.  ScH. — By  an  infinite  quantity  is  not  meant  one  larger  than  any  other, 
or  the  largest  possible  quantity.  It  simply  means  a  quantity  larger  than 
any  assignable  quantity  ;  i.  e. ,  larger  than  any  one  which  has  limits.  The 
mathematical  notion  concerns  rather  the  manner  of  conceiving  the  quantity, 
than  its  absolute  value.  Thus,  a  series  of  Is,  as  1  1  1,  etc.,  repeated  with- 
out stopping,  represents  an  infinite  quantity,  because,  from  the  method  of 
conceiving  the  quantity,  it  is  necessarily  greater  than  any  quantity  which 
we  can  assign  or  mention.  If  we  assign  a  row  of  9s  reaching  around  the 
world,  though  it  is  an  inconceivably  great  number,  it  is  not  as  great  as  a 
series  of  Is  extending  without  limit.  Moreover,  one  infinite  may  be  larger 
than  another  ;   for  a  series  of  2s  extending  without  limit,  as  2  2  2  2,  etc.,  is 


INTEODUCTION.  3 

twice  as  large  as  a  series  of  Is  conceived  in  the  same  way.  It  is  never  of 
any  use  to  try  to  comprehend  the  magnitude  of  an  infinite  quantity  ;  we 
cannot  do  it ;  although  we  can  compare  infinites  just  as  well  as  finites. 

Again,  and- what  is  more  to  our  purpose,  an  infinitesimal  quantity  is  not 
a  quantity  so  small  that  there  can  be  no  smaller.  There  would  be  but  one 
such  quantity  and  hence  no  comparison  of  infinitesimals.  All  that  is  meant 
by  the  term  as  used  in  mathematics  is,  a  quantity  which  is  to  be  treated 
in  the  argument  as  less  than  any  assignable  quantity.  Whether  we  can 
or  cannot  comprehend  its  absolute  magnitude  is  of  no  manner  of  con- 
sequence. Nor  is  absolute  value  usually  of  any  importance  in  pure  mathe- 
matical reasoning.  Thus  2  times  5  is  10  whether  5  be  mites  or  moun- 
tains. In  order  to  free  himself  from  needless  embarrassment  in  the  use 
of  infinitesimals,  the  student  needs  to  keep  constantly  in  mind  the  fact  that, 
In  pure  mathematics,  it  is  the  relation  of  quantities,  rather 
than  their  absolute  values,  with  which  we  are  concerned. 


9»  JPfop, — Any  finite  quantity  divided  by  an  infinite  is  an  infinites- 
imal ;  and  any  finite  quantity  divided  by  an  infinitesimal  is  an  infinite. 

Dem. — Let  a  represent  any  finite  quantity  and  x  any  infinite.     Then  -  is  an  in- 

X 

finitesimal ;  for  the  value  of  a  fraction  depends  upon  the  relative  values  of  its 
numerator  and  denominator,  and  is  less  as  the  ratio  of  numerator  to  denominator 
is  less.     Now,  in  this  case,  a  is  infinitely  less  than  x,  by  the  definition  of  an  infinite. 

Hence  -  is  an  infinitesimal.     Again  -  is  infinite  if  x  is  infinitesimal,  since  a  is  in- 
X  X  . 

finitely  greater  than  cc. 

10,  CoK. — The  reciprocal  of  an  infinite  is  infinitesimal,  and  the  re- 
ciprocal of  an  infinitesimal  is  infinite. 

11,  The  products  of  infinites  by  infinites,  and  of  infinitesimals  by 
infinitesimals  are  denominated  Ot^CTS  :  thus,  if  x  and  y  are  in- 
finites, x%  7/2,  and  coy  are  infinites  of  the  Second  Order ;  if  x,  y, 
and  z  are  infinites,  x^,  z^,  xyz,  x^y,  xy^,  etc.,  are  infinites  of  the  Third 
Order,  The  corresponding  expressions  are  used  with  reference  to 
infinitesimals,  the  product  of  two  infinitesimals  being  caUed  an  infin- 
itesimal of  the  second  order,  of  three,  the  third,  etc. 

12,  ScH. — An  infinite  of  a  lower  order  sustains  a  relation  to  the  next 
higher  similar  to  that  which  a  finite  sustains  to  an  infinite.  Thus  if  x  and  y 
are  infinites,  x-,  xy,  and  y'^  are  infinitely  greater  than  x  and  y.  On  the  other 
hand  if  x  and  y  are  infinitesimals,  x'^,  xy,  and  y^  are  infinitely  less,  and  sus- 
tain a  relation  to  x  and  y,  similar  to  that  which  infinitesimals  sustain  to 
finites. 


INFINITESIMAL  CALCULUS. 


AXIOMS. 

13*  From  expressions  containing  the  sum  or  difference  of  finites 
and  infinites,  the  finites  may  be  dropped  without  affecting  the  ratio. 

14:,  From  expressions  containing  the  sum  or  difference  of  infin- 
ites of  different  orders,  the  terms  containing  the  lower  orders  may  be 
dropped  without  affecting  the  ratio. 

15,  The  order  of  an  infinite  is  not  altered  by  multiplying  or  divid- 
ing it  by  a  finite. 

10»  From  expressions  containing  the  sum  or  difference  of  finites 
and  infinitesimals,  the  infinitesimal  terms  may  be  dropped  without 
affecting  the  ratio. 

17*  From  expressions  containing  the  sum  or  difference  of  infini- 
tesimals of  different  orders  the  terms  containing  the  higher  orders 
may  be  dropped  without  affecting  the  ratio. 

18,  The  order  of  an  infinitesimal  is  not  changed  by  multiplying 
or  dividing  it  by  a  finite. 

Ill's. — Although  the  above  are  conceived  to  be  axioms  in  the  strictest  sense, 
that  is  truths  to  which  the  mind  at  once  assents  as  soon  as  the  terms  used  are 
clearly  comprehended,  the  true  notion  of  infinites  and  intinitesimals  is  so  removed 
from  common  thought  that  a  familiar  illustration  or  two  may  aid  the  comprehen- 
sion. Suppose,  then,  that  the  quantities  under  consideration  were  the  masses  of 
matter  in  the  earth  and  in  the  sun.  If  a  grain  of  sand  were  added  to  or  subtracted 
from  each  or  either  it  would  not  appreciably  affect  the  ratio  of  these  masses.  But 
in  this  instance  the  grain  of  sand  is  by  no  means  infinitesimal  with  reference  to 
either  mass  ;  it  is  &  finite,  though  very  small  part,  of  either  mass. 

Again,  let  x  and  y  be  two  infinite  quantities,  and  a  and  6  two  finite  ones.     There 

can  be  no  difference   between  — ==^    and  -  :   since  to  assume  such  a  difierence 

y  ±zh  y 

would  be  to  assign  some  values  to  a  and  h,  as  respects  x  and  y.     But  by  hypoth- 
esis, the  former  have  no  assignable  values  in  relation  to  the  latter. 

^                                                                                        ..„.,,      ^   a±x       a 
Once  more,  if  a  and  h  are  finite  quantities,  and  x  and  y  infinitesimal,  =  -, 

since  x  and  y  have  no  assignable  values  as  compared  with  a  and  h.     So  also,  x  and 

y  still  being  infinitesimal,  — = —  =  -,  as  x^  and  y^  are  infinitesimals,  (have  no 

y±y'      y 
assignable  values)  with  respect  to  x  and  y. 


INTKODUCTIGN.  5 

ETALUATION   OF   EXPRESSIONS    CONTAINING   EINITES   AND 
INFINITESIMALS,   AND  FINITES  AND  INFINITES, 

Ex.  1.  What  is  the  value  of  the  fraction  - — — -  if  x  is  infinite  and 

dx  -\-  b    ' 

a  and  6  finite  ? 

Solution. — Since  a  and  h  liave  no  assignable  values  in  relation  to  x  they  must 

^x  2 

be  dropped,  and  we  bave  ^.     Now  dividing  both  terms  by  x,  we  have  -  as  the 

value  of  when  re  is  infinite  and  a  and  &  finite. 

'dx  -\-  h 

Ex.  2.  What  is  the  value  of  the  fraction  in  the  last  example  if  x  is 
infinitesimal  and  a  and  h  finite  ? 

Solution. — As  x  is  infinitesimal  ^x  and  3ic  are  also  infinitesimal,  and  hence  have 
no  value  in  relation  to  a  and  h,  and  must  be  dropped.     Hence  the  value  of  the 

fraction  is  —  -. 

0 

Ex.  3.  What  is  the  value  of  -— — when  x  is  infinite  ?     When 

X  is  infinitesimal  ? 

Ans.,  When  x  is  infinite,  6  ;  when  infinitesimal,  3. 

^x 

X 

Ex.  4.  What  is  the  value  of  y  in  the  equation  y  = when  x 

--\-x 

X 

is  infinite  ?     When  x  is  infinitesimal  ? 

Atis.j  When  x  is  infinite,  —  5  ;  when  infinitesimal,  -. 

Ex.  5.  What  is  the  value  of  y  in  the  expression  y  =  — - —  when  x 

JL  "j~  X 

is  infinite  ?     When  x  is  infinitesimal  ? 

Ans.,  When  x  is  infinite,  0  ;  when  infinitesimal,  1. 

-^      ^    -^x,    .  .    .-,         ,         .    ax^  -\- hx"^  -^  ex  -\r  d      .  ••/»..« 

Ex.  6.  What  IS  the  value  of when  x  is  mnmte? 

m^3  _|_  ifirjQ'i  -\-  px  4-  q 

When  X  is  infinitesimal  ? 

Ans.,  WTien  x  is  infinite,  —  ;  when  infinitesimal,  -. 

m  q 

2x^  —  5m^x 
Ex.  7.  What  is  the  value  of  y  m  the  expression  y  =  — • 

ox    •        TTIX 

when  X  is  infinite  ?     When  x  is  infinitesimal  ? 

Ans.,  When  x  is  infinite,  0  ;  when  infinitesimal,  5m, 


6  INFINITESIMAL  CALCULUS. 

•  doc'^  A-  2  7*^ 1 

Ex.  8.  "What  is  the  value  of  y  in  the  equation  y  = - 

when  X  is  infinite  ?     "When  x  is  infinitesimal  ? 

Ans.,  When  x  is  infinite,  y=Qo',  when  infinitesimal,  y  =  ■ —  \. 

?)X 

Ex.  9.  When  x  and  v  are  infinitesimals  what  is  the  value  of  *--  ? 

^/zs.,  We  cannot  tell ;  as  we  know  nothing  about  the  relation  be- 
tween X  and  y. 

3x 

Ex.  10.  What  is  the  value  of  —  when  y^  =  9x  and  x  and  y  are 

infinite  ?  Ani^.,  oo. 

Ex.  11.  Same  as  Ex.  10,  only  x  and  y  infinitesimal?  Ans.,  0. 

Ex.  12.  What  is  the  value  of  y  in  the  equation  y^  =  —- ,  when 

X  is  infinitesimal  ?  Ans.,  0. 


CONSTANTS  AND  TARIABLES. 

19.  A.  CottStatlt  quantity  is  one  which  maintains  the  same 
value  throughout  the  same  discussion,  and  is  represented  in  the  no- 
tation by  one  of  the  leading  letters  of  the  alphabet. 

20*  VuTiable  quantities  are  such  as  may  assume  in  the  same 
discussion  any  value,  within  certain  limits  determined  by  the  nature 
of  the  problem,  and  are  represented  by  the  final  letters  of  the 
alphabet. 

2JL,  CoR.  — Any  exjjression  containing  a  variable  is,  when  taken  as  a 

whole,  a  variable.     Thus  the  value  of  the  entire  exjjression  (4a — 3^^  -[-5)2 
varies  if  x  varies  ;  so  that  taken  as  a  whole  it  is  a  variable. 

[Note. — These  notions  should  be  already  familiar  from  General  Geometry,  page  9,  and  are  in- 
troduced here  only  to  give  completeness,  and  for  review.] 

22,  Variables  are  distinguished  as  Independent  and  Dependent. 

23.  An  Independent  Variable  is  one  to  which  we  assign 
arbitrary  values,  or  upon  whose  law  of  variation  we  make  some  arbi- 
trary hypothesis. 

24:,  A-  Dependent  Yariahle  is  one  which  varies  in  value  in 
consequence  of  the  variation  of  the  independent  variable  or  vari- 
ables. 

IiiL. — ^ThuB,  in  the  equation  of  the  parabola,  y'^  =  2px,  if  we  assign  arbitrary 


INTRODUCTION. 


values  to  x  and  find  tlie  corresponding  values  of  y,  we  make 
X  the  independent  variable,  and  p  the  dependent  variable. 
Again,  and  what  is  more  to  our  present  purpose,  if  we  as- 
sume X  to  vary  in  some  particular  way,  as  by  taking  on  equal 
increments,  as  DD',  D'D",  D"D"',  etc.,  2/ will  evi- 
dently vary  ia  some  other  way,  but  still  in  a  way  depending 
upon  the  way  in  which  x  varies,  and  upon  the  nature  of  the 
curve,  or,  what  is  the  same  thing,  upon  the  form  of  the 
equation  of  the  curve.  In  this  case  also,  x  is  the  independ- 
ent and  7/  the  dependent  variable. 


Fig.  4. 


ScH. — This  distinction  is  made  simply  for  convenience,  and  is  not  founded 
in  any  difference  in  the  nature  of  the  variables ;  either  variable  may  be 
treated  as  the  independent  variable. 

2S,  A.n  JSquicresceflt  variable  is  one  which  is  assumed  to  in- 
crease or  decrease  by  equal  increments  or  decrements,  as  x  in  the  last 
illustration. 

26.  Contemporaneous  Tncrefnents  are 

such  as  are  generated  at  the  same  time. 


iLii. — Thus  let^=/(cc)  represent  the  equation  of  AM  in 
the  figure.  Suppose  we  contemplate  the  values  of  x  and  y 
at  the  point  P'.  Now  if  ic  takes  the  increment  D'D",  y 
takes  the  contemporaneous  increment  P"E'.  So  also  we 
see  that  DD',  P'E,  and  PP'  are  contemporaneous  incre- 
ments of  the  abscissa,  ordinate,  and  arc,  respectively. 


Fig.  5. 


FUNCTIONS   AND    THEIR   FORMS, 

27*  JL  Function  is  a  quantity,  or  a  mathematical  expression, 
conceived  as  depending  for  its  value  upon  some  other  quantity  or 
quantities. 

III.  — A  man's  wages  for  a  given  time  is  a  function  of  the  amount  received  per 
day ;  or,  in  general,  his  wages  is  a  function  of  both  the  time  of  service  and  the 
amount  received  per  day.  Again,  in  the  expressions  y  =  2ax-,  y  =  x^  —  26x  +  5, 
2/  =  2  log  ax,  y  =  a'',  y  is,  a  function  of  x  ;  since,  the  numbers  2,  5,  a  and  &  being 
considered  constant,  the  value  of  y  depends  upon  the  value  we  assign  to  x.     For 

a  like  reason  \/a'^  —  x"^,   and  3aa;2  —  2\/a;  may  be  spoken  of  as  functions  of  x. 
Once  more,  the  ordinate  of  a  curve  is  a  function  of  the  abscissa. 

ScH. — There  is  a  sense  in  which  the  dependent  variable  (or  function)  is  a 
function  of  the  constants  as  well  as  of  the  variable  or  variables  which  enter 
into  its  value.  So  also  it  is  a  function  of  the  form  of  the  expression,  that 
is,  its  value  depends  in  part  upon  the  form  of  the  expression  as  well  as 
upon  the  value  of  the  independent  variable.     Thus  if  we  have  y  =  a\ogx 


8  INFINITESIMAL   CALCULUS. 

-f-  h,  and  y  =  x^  —  ex,  though  in  each  case  ?/  is  a  function  of  x,  speaking 
according  to  the  definition,  nevertheless  it  is  not  the  same  function  in  both 
cases.  Its  value  depends  upon  the  value  of  x,  upon  the  constants,  and 
upon  the  form  of  the  expression  involving  these  quantities.  But  the  con- 
ception expressed  in  the  definition  is  the  ordinary  one. 

28*  Functions  are  classified  by  their  forms  as  Algebraic  and 
TraTiscendentalf  and  the  latter  are  subdivided  into  Trigono- 
metrical and  Circular^  Logarithmic  and  Exponential. 

29,  An  Algebraic  Function  is  one  which  involves  only  the 
elementary  methods  of  combination,  viz.,  addition,  subtraction,  mul- 
tiplication, division,  involution  and  evolution.  Thus  in  y=  ax^ — 3^", 
y  is  an  algebraic  function  of  x. 

30,  A  Trigonometrical  Function  is  one  which  involves 
sines,  cosines,  tangents,  cotangents,  etc.,  as  variables  ;  thus  ?/=:sinar, 
y  =  sin  X  tan  x,  etc. 

31,  A  Circular  Function  is  one  in  which  the  concept  is  a 
variable  arc  (in  the  trigonometrical  the  concept  is  a  right  line).  These 
are  written  thus  :  y  =  sin~^^,  read  "  y  equals  the  arc  whose  sine  is  a; "; 
y  ==  isiii'^x,  read  " y  equals  the  arc  whose  tangent  is  x." 

III. — Notice  that  in  the  expression  2/ =  tan— i  .-j;,  it  is  the  arc  which  we  are  to 
think  of,  while  in  the  expression  x  =  tan  y  it  is  the  tangent,  which  is  a  right  line. 
Trigonometrical  functions  are  right  lines  ;  circular  functions  are  arcs.  These 
functions  are  mutually  convertible  into  each  other  ;  thus  y  =  sin—'  x  is  equivalent 
to  a;  =  sin  y,  the  only  difference  being  that  in  the  former  we  think  of  the  arc,  the 
sine  being  given  to  tell  what  arc,  and  in  the  latter,  we  think  of  its  sine,  the  arc 
being  given  to  tell  what  sine. 

The  circular  functions  y  =  sin~^^,  y  =  cosr'^x^  y  =  sec~^x,  etc.,  are 
often  called  Inverse  Trigonometrical  Functions. 

32,  A  Logarithmic  Function  is  one  which  involves  loga- 
rithms of  the  variable  ;  as  y  =  log  x,  log^  2/  =  3  log  ax,  etc. 

33,  An  Fxponential  Function  is  one  in  which  the  vari- 
able occurs  as  an  exponent ;  as  y  =  a"",  z  =  x'-',  etc. 


34,  Functions  are  further  distinguished  as  Explicit  and  Im- 
plicit, 

35,  An  Explicit^  Function  is  a  variable  whose  value  is  ex- 
pressed in  terms  of  another  variable  or  other  variables  and  constants. 

Thus  in  y  =  2ax^  —  3^"^,  y  is  an  explicit  function  of  x. 

*  From  explicitum,  unfolded.     The  function  is  disentangled  from  the  other  quantities. 


INTRODUCTION.  9 

36,  An  Implicit^  Fmiction  is  a  variable  involved  in  an 
equation  which  is  not  solved.  Thus  in  x"^  —  ^xy  -f  2?/  =  16,  ?/  is  an 
implicit  function  of  x,  or  x  is  an  implicit  function  of  y.  When  we 
can  solve  the  equation,  an  implicit  function  may  always  be  expressed 
as  exphcit. 

37*  dotation.  When  we  wish  to  write  that  y  is  an  explicit 
function  of  x,  and  do  not  care  to  say  precisely  what  the  form  of  the 
function  is,  we  write  y  =f(x),  read  "i/  =  a  function  of  x."  If  we 
wish  to  indicate  several  different  forms  of  dependence  in  the  same 
discussion,  we  use  other  letters,  as  2/=/(^),  y=F{x),  y=  cp{x),  etc., 
or  use  subscripts  or  accents  as  y  =f(x),  y  ==zf'(^x),  etc.  Such 
symbols  are  read  "y  =  the/",  large  F,  cp,  f  sub-one,  f  prime,  etc., 
function  of  x"  as  the  case  may  be. 

When  we  wish  to  write  that  x  and  y  are  functions  of  each  other,  or 
that  y  is  an  implicit  function  of  x,  or  x  an  implicit  function  of  y, 
without  being  more  specific,  we  write  F{x,  y)  =  0,  or  f{x,  y)  ==  0, 
or  (p{x,  y)  r=  0,  etc.  ;  and  read  "function  x  and  y  =  0,"  the  F  func- 
tion X  and  2/  =  0,  etc.  This  form  symbolizes  any  equation  between 
two  variables  with  all  the  terms  transposed  to  the  first  member. 


3S,  Again,  functions  are  distinguished  as  Incveasitig  and 
Decreasing, ' 

30*  A.n  Increasing  Function  is  a  function  that  increases 
as  its  variable  increases,  and  decreases  as  its  variable  decreases. 

4:0,  Jl  Decreasing  Function  is  a  function  which  decreases 
as  its  variable  increases,  and  increases  as  its  variable  decreases. 

III,. — In  the  expressions  t/-  =  'Ipx,  y  =  log  'X,,  y  t^  a^,  y  is  an  increasing  function 
of  X.  In  the  expressions  y  = —,  y"-  -\-  x^  =  E-,  y  =  log  — ,  y  is  a  decreasing  func- 
tion of  X.  For  what  vahies  of  cc  is  2/  an  increasing  function  of  its  variable,  and 
for  what  a  decreasing,  in  the  following  :    y^  =  ax^  —  x'^,  y  =  sin  a;,  y  =z  cosa;  ? 


41,  OOhe  Infinitesimal  Calculus  treats  of  Continuous 
Number^  and  is  chiefly  occupied  in  deducing  the  relations  of  the  con- 
temporaneous infinitesimal  elements  of  such  number  from  given  re- 
lations between  finite  values,  and  the  converse  process,  and  also  in 
pointing  out  the  nature  of  such  infinitesimals  and  the  methods  of 
using  them  in  mathematical  investigation. 


*  From  implicituiti,  infolded,  entangled. 


10 


INFINITESIMAL  CALCULUS. 


III. — Let  y^  =  8x  be  the  equation  of  the  parabola  in  the 

figure.     Here  we  have  the  relation  between  finite  values  of 

y  and  x  expressed.     Now  suppose  x  takes  an  infinitesimal 

increment  as.  D  D '  *,  what  increment  does  y  take  ?     The  cal- 

.    4 
cuius  shows  us  that  the  increment  which  y  takes  is  -  times 

y 

as  large  as  the  increment  which  x  takes  ;  that  is,  it  shows  us 
the  relation  between  the  elements  of  the  variables  y  and  x, 
when  we  know  the  relation  between  finite  values.  This  is 
the  province  of  the  Differential  Calculus,     The  converse  of 


Fig.  6. 


this  problem  is,  What  is  the  equation  of  the  curve  whose  ordinate  varies  -  times 

as  fast  as  its  abscissa  ?  that  is,  having  given  the  relation  between  the  infinitesimal 
elements  of  y  and  x,  to  find  the  relation  between  finite  values.  This  is  the  prov- 
ince of  The  Integral  Calculus. 

4:2,  There  are  two  branches  of  the   Calculus,  yiz.,  XTie  Differ- 
ential Calculus^  and  The  Integral  Calculus, 


*  Of  course  all  such  attempts  to  represent  infinitesimals  to  tlie  eye,  are  egregious  exaggerations; 
nevertheless  they  are  of  great  service  to  the  roiad. 


THE 


INFINITESIMAL   CALCULUS. 


CHAPTER  I. 

THE  JDIFFEBENTIAL    CALCULUS. 


SECTION  L 
DifFerentiation  of  Algebraic  Functions. 

4:8,  The  DiffereTitial  Calculus  is  that  branch  of  the  Infin- 
itesimal Calculus  which  treats  of  the  methods  of  deducing  the 
relations  between  the  contemporaneous  infinitesimal  elements  of  vari- 
ables, from  given  relations  between  finite  values. 

4:4:,  A.  Diffevential  is  the  difference  between  two  consecutive 
states  of  a  function,  or  variable.     It  is  the  same  as  an  infinitesimal. 

45,  Consecutive  Values  of  a  function  or  variable  are  values 
which  differ  from  each  other  by  less  than  any  assignable  quantity. 

Consecutive  Points  on  a  line  are  points  nearer  to  each  other  than 
any  assignable  distance. 

III. — Suppose  y  =  2x^ —  Sx.  Now  let  x  be  supposed  to  increase  infinitesimally, 
2/  will  also  change  infinitesimally.  Call  the  new  value  of  y,  y\  Then  y'  =2x'^  — 
3x'.  In  such  a  case  a;  and  x'  are  consecutive  values  of  the  variable,  and  y  and  y' 
are  consecutive  values  of  the  function.  But  by  this  we  do  not  mean  that  x  and  x' 
{or  J  and  y' )  are  so  nearly  equal  thai  there  can  he  no  intermediate  value,  for  this  would 
be  to  make  an  infinitesimal  mean  a  quantity  so  small  that  there  can  be  no  smaller, 
which  is  not  its  meaning  as  used  in  mathematics  (7).  AH  that  is  meant  by  saying 
that  y  and  y'  are  consecutive  values  is  that  they  are  to  he  reasoned  upon  as  having 
no  assignable  difference. 

So  also  in  speaking  of  consecutive  points  on  a  line,  as  D  and  D',  or  P  and  P', 
Mg.  6,  we  do  not  conceive  them  as  actually  in  juxtaposition  ;  but  we  mean  simply 
that  we  are  to  reason  upon  them  as  nearer  each  other  than  any  assignable  distance. 

46,  Wotatiofl,  The  differential  of  a  variable  (one  of  its  infini- 
tesimal elements)  is  represented  by  writing  the  letter  d  before  it. 


12  THE  DIFFEKENTIAL  CALCULUS. 

Thus,  doc,  read,  "  differential  x"   Of  course  the  letter  d  is  not  to  be  con- 
founded with  a  factor  ;  it  is  simply  an  abbreviation  for  differential. 

[Caution. — The  student  should  be  careful  and  not  allow  himself  to  read  such 
expressions  as  dy,  dx,  etc. ,  by  merely  naming  the  letters  as  he  would  ay,  ax,  etc. 
The  former  should  always  be  read  "  differential  y,"  "differential  x,"  etc.] 


RULES   FOR   DIFFERENTIATING   ALGEBRAIC   FUNCTIONS. 

47.    BULE   1. To     DIFFERENTIATE    A    SINGLE    YAEIABLE     SIMPLY    WHITE 

THE  LETTER  d  BEFORE  IT. 

Dem. — Let  us  take  the  function  y  =z  x.  The  consecutive  state  of  the  variable 
is  ic  -|-  dx.  Now  representing  the  change  in  y  which  is  produced  by  this  change 
in  X  by  cZy  {dx  and  dy  being  the  contemporaneous  increments  of  the  variable  and 
the  function),  we  have 

1st  state  of  the  function,  y  =  x, 

2nd,  or  consecutive  state, y  -\-  dy  ^x  -\-  d^. 

Subtracting  the  1st  from  the  2nd,   dy  =  dx,  which 

being  the  difference  between  two  consecutive  states  of  the  function  is  its  differen- 
tial {4:4:).       Q.  E.  D. 

ScH. — This  rule  is  evidently  only  the  same  thing  as  the  notation  requires, 
and  its  formal  demonstration  would  be  unnecessary  except  for  the  purpose 
of  uniformity  in  treating  the  several  cases  of  differentiation. 

III. — Let  M  N  be  a  line  passing  through  the  origin  and  making  an  angle  of 
450  with  the  axis  of  x.  Its  equation  is  ?/  =  cc.  Let  P  be  any  point  in  the  line, 
AD  =  ic,  and  P D  =  ?/.  Let  a;  take  the  infinitesimal  increment  D  D ' {dx),  then 
y  becomes  P'  D'.     Now  the  first  state  of  the  function  is 

P D  =  A  D,  or  2/ =  .T, 
The  second  or  consecutive  state  is  PD  +  P'E  =  AD+  DP",  ""r  y^dy  =  x-\-dx. 
Subtracting  we  have  P'  E  ^  D  D',  or  dy  =  dx. 

Now  that  the  increment  of  y  (or  dy)  is  equal  to  the  in-  Y  /jyi 

crement  of  x  (or  dx)  in  this  case  is  readily  seen  from 
the  figure  ;  for,  as  P'PE  =  45°,  P'E  =  PE,  or  DD'. 
dy  =  dx,  then,  means  that  the  contemporaneous  incre- 
ments of  X  and  y  are  equal,  or  that  x  and  y  increase  at 
the  same  rate. 


DDT  X 


Fig.  7. 

48,  B  ULE  2. — Constant  factors  or  divisors  appear  in  the  differ- 
ential THE  SAME  AS  IN  THE  FUNCTION. 

Dem.  — Let  us  take  the  function  y  =  ax,  in  which  a  is  any  constant,  integral  or 
fractional.  Let  .v  take  an  infinitesimal  increment  and  become  x  -\-  dx  ;  and  let  dy 
be  the  contemporaneous  increment  of  2/,  so  that  when  x  becomes  x  -\-  dx,  y  be- 
comes y  -|-  dy.     We  then  have 


DIFFERENTIATION   OF  ALGEBRAIC   FUNCTIONS. 


13 


1st  state  of  tlie  function, y  ■=  ax\ 

2nd,  or  consecutive  state, y  -\-  dy  =  a'x  -\-  dx) 

Subtracting  the  1st  from  the  2nd, 


a.x  -j-  adx: 


dy  =  adx,  ■ 

■which  being  the  difference  between  two  consecutive  states  of  the  function  is  its 
differential  {44:) .  Now  the  factor  a  appears  in  this  differential  just  as  it  was  in 
the  function,     q.  e.  d. 

Iljl. — Let  y  =z  ax  he  the  equation  of  the  line  MN. 
PD  and  P'D'  representing  consecutive  ordinates,  DD' 
represents  dx,  and  P'  E  represents  dy.  Here  it  is  evident 
that  P'E  =a  X  DD';  for  from  the  triangle  PP'E  we 
have  P'E  =  tanP'PE  X  PE.  But  tan  P'PE  = 
tan  MAX  =  a.  Th'e  meaning  in  this  case  is,  therefore, 
that  the  ordinate  increases  a  times  as  fast  as  the  abscissa.  ^^^   °- 

If  «  :■=  1,  or  tan  45°,  the  ordinate  and  abscissa  increase  at  equal  rates  ;  if  a  <^  1, 
i.  e. ,  if  the  angle  is  less  than  45°  the  ordinate  increases  more  slowly  than  the 
abscissa  ;  if  «  ^  1,  the  ordinate  increases  more  rapidly  than  the  abscissa. 

40,  RULE  3. — Constant  terms  disappear  in  differentiating:    or 

THE  DIFFERENTIAL  OF  A  CONSTANT  IS  0 . 

Dem. — Let  US  take  the  function  y  =  ax  -±1  h,  in  which  a  and  h  are  constants. 
Let  x  take  an  infinitesimal  increment  and  become  cc  -\-  dx  ;  and  let  dy  be  the  con- 
temporaneous increment  of  y,  so  that  when  x  becomes  x  -\-  dx,  y  becomes  y  -\-  dy. 
We  then  have 

1st  state  of  the  function, y  =  ax  ±h  \ 

2nd,  or  consecutive  state, y  -\-  dy  =  a(x  +  (^^)  rt  &, 

or y  -\-  dy  =  ax  -\-  adx  =fc  h. 

Subtracting  the  1st  state  from  the  2nd,  dy  =^  adx,  which 

being  the  difference  between  two  consecutive  states  of  the  function  is  its  differen- 
tial {44).  Now  from  this  differential  the  constant  term  ±  6  has  disappeared. 
We  may  also  say  that  as  a  constant  retains  the  same  value  there  is  no  difference 
between  its  consecutive  states  (properly  it  has  no  consecutive  states).  Hence  the 
differential  of  a  constant  may  be  spoken  of  (though  with  some  latitude)  as  0. 

Q.  E.  D. 

III.— Let  y==ax-\-hhe  the  equation  of  the  line  M  N. 
Now  the  relative  rates  of  increase  of  the  abscissa  and  or- 
dinate, that  is  the  relation  of  dy  to  dx,  is  evidently  not 
affected  by  h  which  is  A  B  ;  for,  if  we  were  to  draw  a  line 
through  the  origin  parallel  to  M  N ,  the  contemporane- 
ous increments  of  its  co-ordinates  would  be  the  same  as 
those  of  M  N .  Again,  we  can  see  that  the  constant  term 
does  not  affect  the  differential,  i.  e.,  the  difference  between 

the  consecutive  states  of  y,  by  observing  that  these  two  states  are  represented  by 
PD  and  P'  D',  each  of  which  contains  the  constant  as  a  part  of  it,  whence  the 
difference  between  them  is  not  affected  by  it. 

SO,  Cor. — An  infinite  variety  of  functions  differing  from  each  other 
only  in  their  constant  terms  still  have  the  same  differential. 


Fig.  9. 


14  THE  DIFFEEEKTIAL  CALCULUS. 

31,  RULE  4. — To  differentiate  the  algebraic  sum  of  several  va- 
riables, DIFFERENTIATE  EACH  TERM  SEPARATELY  AND  CONNECT  THE  DIFFEREN- 
TIALS WITH  THE  SAME  SIGNS  AS  THE  TERMS. 

Dem. — Let  u  =  x  -{-  y  —  z,  u  representing  the  algebraic  sum  of  the  variables 
X,  y,  and  —  z.  Then  is  the  differential  of  this  sum  or  du=.dx  -\-  dy  —  dz.  For 
let  dx,  dy,  and  dz  be  infinitesimal  increments  of  x,  y,  and  z  ;  and  let  du  be  the  in- 
crement which  u  takes  in  consequence  of  the  infinitesimal  changes  in  x,  y,  and  z. 
We  then  have 

1st  state  of  the  function, m  =  £c  +  2/  —  ^  ■> 

2nd,  or  consecutiye  state, u  -\-  du  =  x  -\-  dx  -\- y  -\-  dy  —  {z  -\-  dz), 

or u  -^  du  :=  X  -}-  dx  -{-  y  -^  dy  —  z  —  dz. 

Subtracting  the  1st  state  from  the  2nd,  du  =  dx  -{-  dy  —  dz.  q.  e.  d. 

III. — "We  may  illustrate  this  by  conceiving  x  and  y  to  be  forces  acting  to  raise 
a  weight,  and  z  a  force  acting  to  prevent  the  raising,  u  being  the  aggregate  effect  of 
all,  i.  e.  their  algebraic  sum  (Complete  Algebra,  65).  Now  if  x,  y,  and  z  each  re- 
ceive an  infinitesimal  increment,  which  we  will  call  respectively  dx,  dy,  and  dz,  it 
is  evident  that  the  increment  of  lifting  force  is  dx  -\-  dy,  and  as  the  increment  of 
the  depressing  force  is  dz,  the  combined  effect  of  the  change  is  dx  -\-  dy  —  dz, 
which  is  the  change  in  u.  Moreover,  since  this  quantity  dx  -\-  dy  —  dz  is  the  ag- 
gregate of  a  finite  number  of  infinitesimals,  it  must  be  itself  infinitesimal.  Hence 
the  change  in  u  is  infinitesimal,  or  du. 

ScH. — It  is  important  to  notice  that  the  above  reasoning  is  entirely  inde- 
pendent of  the  relative  values  of  the  infinitesimals  dx,  dy,  and  dz.  These 
may  be  conceived  as  equal,  or  as  sustaining  any  finite  ratio  whatever  to 
each  other,  only  so  that  they  remain  infinitesimal. 

32,  RULE  5. — The  differential  of  the  product  of  two  variables 

15  the  differential  of  the  first  into  the  second,  plus  the  differential 

OF  THE  SECOND  INTO  THE  FIRST. 

Dem, — Letw  =^  xy  be  the  first  state.  The  consecutive  state  is  w  -f-  dw  = 
{x  -\-  dx){y  -\-  dy)  =xy  -\-  ydx  -\-  xdy  -\-  dxdy.  Subtracting  the  1st  state  from  the 
2nd,  or  consecutive  state,  we  have  dxL  =  ydx  -f-  ^^y  +  <^^'  •  ^V-  Now  ydx  and  xdy 
are  infinitesimals  of  the  1st  order,  and  dx  •  dy,  being  the  product  of  two  infinitesi- 
mals, is  of  the  2nd  order  and  must  be  dropped  (17).     Therefore  du  =  ydx  -f-  xdy. 

Q.  E.  D. 


III. — Let  u  represent  the  area  of  the  rectangle  A  BC D,  £C  =     ^ e 


C 


*  A  B,  and  y  =  *BC.     Then  u  =  xy.     Let  B&  represent  dx,  and     p  - 

Oc",dy.     Whence  ShCc'  =  *ydx,  DdCc"=*xdy,  Cc'cc"  ==* 

dx  '  dy,   and  du  =  *BhCc'  +  DdOc"  -\-  Cc'cc".      Now  since 

cc'  is  infinitesimal  and  c'h  is  finite,  Cc"cc"  is  infinitesimal  with     A  B 

reference  to   B&Cc',  as  for  a  like  reason  it  is  with  reference  to  Fig.  10. 

DdCc"  ;  hence  it  is  to  be  omitted  as  having  no  assignable  value  with  reference  to 

them. 

Another  view  which  may  be  taken  of  this  is  to  consider  that  it  is  the  rate  at  which 

*  In  such  cases  =  signifies  "  is  represented  by,"  and  is  used  for  brevity. 


DIFFEBENTIATION  OP  ALGEBRAIC  FUNCTIONS.  15 

the  rectangle  is  increasimj  when  a;  =  A  B  and  1/  =  BC,  not  the  amount  of  change 
in  the  area  after  x  and  y  shall  have  increased  more  or  less  :  in  other  words,  we  seek 
for  the  difference  between  consecutive  values  of  the  area.  Now  it  is  easy  to  see 
that  the  rale  at  which  the  rectangle  A  BCD  starts  to  increase,  depends  upon  the 
length  of  the  side  BC  iy)  and  the  rate  at  which  it  starts  to  move  to  the  right,  -j-  the 
length  of  DC  (x)  and  the  rate  at  which  it  starts  to  move  upward.  Letting  dx 
represent  the  rate  at  which  A  B  starts  to  increase  (by  being  the  amount  which  it 
would  increase  in  an  infinitesimal  of  time),  and  dy  represent  in  like  manner  the 
rate  at  which  y  starts  to  increase,  we  readily  see  that  du  =  ydx  -f-  scdy  is  the  rate 
at  which  the  area  starts  to  increase.  Moreover,  we  see  that  this  is  equally  true 
whether  dy  =^  dx,  or  whether  one  is  any  finite  multiple  of  the  other  ;  all  that  is 
necessary  being  that  both  be  infinitesimals  of  the  same  order. 

^3,  B  ULE  6. — The  differential  of  the  product  of  several  varia- 
bles IS  THE  SUM  OF  THE  PRODUCTS  OF  THE  DIFFERENTIAL  OF  EACH  INTO  THE 
PRODUCT  OF  ALL  THE  OTHERS. 

Dem. — Let  u  :=  xyz  ;  then  du  =  yzdx  -f-  xzdy  -f-  xydz. 

For  the  1st  state  of  function  is u  =  xyz, 

2nd,  or  consecutive  state, u-\-duz=  {x-\-dx){y-\-dy){z-\-dz), 

or u-{-du  ■=  xyz  -\-  yzdx  -f-  xzdy  -\-  xydz  -f-  xdydz  -f-  ydxdz  -\-  zdxdy  -\-  dxdydz. 

Subtracting  and  dropping  infinitesimals  of  higher  orders  than  the  first  we  have 
du  =-  yzdx  -f-  xzdy  -\-  xydz. 

In  a  similar  manner  the  rule  can  be  demonstrated  for  any  number  of  variables. 

Q.  E.  D. 

S4:,  RULE  7. — The  differential  of  a  fraction  having  a  variable 

NUMERATOR  AND  DENOMINATOR  IS  THE  DIFFERENTIAL  OF  THE  NUMERATOR 
MULTIPLIED  BY  THE  DENOMINATOR,  MINUS  THE  DIFFERENTIAL  OF  THE  DENOMI- 
NATOR MULTIPLIED  BY  THE  NUMERATOR,  DIVIDED  BY  THE  SQUARE  OF  THE 
DENOMINATOR. 

Dem. — Let  u  =  -  ;  then  is  du  =  — ^ — - — -.  For  clearing  of  fractions  yu  =  x. 
Differentiating  this  by  Kul^  5,  udy  -\-  ydu  =  dx.  Substituting  for  u  its  value,  we 
have  ' — : — f-  ydu  =  dx.      Finding  the  value  of  du,  we  have  du  =  - — '■ — -. 

y  y 

Q.  E.  D. 

SS»  CoR. — The  differential  of  a  fraction  hamng  a  constant  numerator 
and  a  variable  denominator  is  the  product  of  the  numerator  with  its  sign 
changed  into  the  differential  of  the  denominator,  divided  by  the  square  of  ' 
the  denominator. 

Dem. — Let  u  =  -.  Differentiating  this  by  the  rule  and  calling  the  differential 
of  the  constant  (a),  0,  we  have  du  = = ~.     o.  e.  d. 


16  THE  DIFFERENTIAL   CALCULUS. 

ScH. — ^If  the  numerator  is  variable  and  the  denominator  constant  it  falls 
under  Rule  2. 

S6.  RULE  8. — The  differential  of  a  vaeiable  affected  with  an 

EXPONENT  IS  THE  CONTINUED  PEODUCT  OF  THE  EXPONENT,  THE  VARIABLE  WITH 
ITS  EXPONENT  DIMINISHED  BY  1,  AND  THE  DIFFERENTIAL  OF  THE  VARIABLE. 

Bem. — 1st.  When  the  exponent  is  a  positive  integer. — Let  y  =  x"^,  m  being  a  pos- 
itive integer  ;  then  dy  =  mx^~^dx.  For  y  z=  x^  ::=  x  •  x  •  x  '  xio  m  factors.  Now 
differentiating  this  by  Bute  6,  we  have 

dy  =  {XXX  to  m  —  1  factors)  dx  -f-  {xxx  to  m  —  1  factors)  dx  +  etc.,  to  m  terms, 
or  dy  =  ic"»-^dx  +  x'^—^dx  -\-  x"'—^dx  -f-  etc.,  to  m  terms. 
.  • .  dy  =:  mx'^—^dx. 

-   m      . 
2nd.    When  the  exponent  is  a  positive  fraction. — Let  y  =z  xn,  —  being  a  positive 

TO  !^  — 1 

fraction  ;   then  dy  =  — X"     dx.     For  involving  both  members  to  the  nth  power 

n 

we  have  2/"  =  ^C".      Differentiating  as  just  shown,  ny"~^dy  =  mx^—^dx.      Now 

m  mn — m 

from  y  =  a  «,  we  have  y"—^  =  x~i    .     Substituting  this  in  the  last  form,  we  have 

mn — m  »j.  mv — m  ™    in 

nx    n~dy  =  mx'^—^dx,  or  dv  =  — x^  Ti    dx  =  —  X"~  dx. 

^  n  n 

3rd.  When  the  exponent  is  negative. — Let  y  =  x—",  n  being  integral  or  fractional ; 
then  dy= — nx-^—^dx.     For  i/ =  x—"  =  — ,  which  differentiated  by  jRwte  7,  Cor., 

gives  dy=: ;; — '—  =  —  nxr-"—^dx.      All  three  of  which  forms  agree  with  the 

enunciation  of  the  rule.     q.  e.  d. 

S7*  Cor. — The  differential  of  the  square  root  of  a  variable  is  the  dif- 
ferential of  the  variable  divided  by  twice  the  square  root  of  the  variable. 

Dem. — Let  y  =  \/x  =  x  .  Differentiating  by  the  rule  we  have  dy  =-ix^  dx=. 
1  -2 ,  dx 

iX      dX  =  =.       Q.  E.  D. 

2v/x 

ScH. — Special  rules  can  be  readily  made  for  other  roots,  but  it  is  un- 
necessary. The  square  root  is  of  such  frequent  occurrence  as  to  make  the 
special  process  expedient.  Of  course  the  general  rule  can  always  be  used, 
if  desired. 


EXERCISES. 

[Note. — The  following  examples  are  designed  to  give  practical  skill  in  applying  tlie  rules  for 
differentiating  algebraic  functions.  The  student  should  not  advance  beyond  these,  till  he  has 
the  rules  firmly  fixed  in  memory,  and  can  apply  them  with  facility  to  all  forms  of  algebraic  func- 
tions.] 

Ex.  1 .  Differentiate  y  =  Qx  —  4.  dy  =  6dx. 

QuEKT. — What  three  iiales  apply?  Be  careful  to  repeat  the  rules  in  applying 
them  to  the  solution  of  the  examples,  and  thus  render  them  familiar. 


DIFFEBENTIATION  OF  ALGEBRAIC  FUNCTIONS.  17 

-Ex.  2.  Differentiate  y  =  a''  +  da'x-'  +  Sa'x^  +  x^ 

Solution. — The  differential  of  i/ is  d^/-  [Bepeat  Rule  1.]  To  differentiate  the 
second  member  we  notice  1st,  that  it  consists  of  several  terms,  and  hence  proceed 
to  differentiate  each  term  separately.  [Repeat  liule  4.]  a^  being  a  constant  term, 
disappears.  [Eepeat  Eule  3.]  To  differentiate  Sa^x^,  we  notice  l«t  that  the  con- 
stant factor  3a4  will  be  a  factor  in  the  differential.  [Repeat  Eule  2.]  The  differ- 
ential of  a;2  is  2xdx.  [Repeat  Bule  8.]  Hence  the  differential  of  Sa'^x'^  is  GalTc/a?. 
[In  like  manner  proceed  with  the  other  terms,  giviiig  the  reason  for  each  step  hy 
repeating  the  appropriate  rule.^ 

Ex.  3 .  Differentiate  u  =  2ax  —  3^^  _|_  ahx^  —  5. 

Result,  du  =  (2a  —  6x  -{-  dabx^)dx. 

Ex.  4.  Differentiate  y  =  ^x^  — ■  2x  —  Bin. 

Ex.  5.  Differentiate  u  =  ab  — ■  6x^  -{-  2ax. 

Ex.  6.  Differentiate  it  =  ax-y\ 

QuEBiES. — What  is  the  most  general  feature  of  the  function  ax^^?  What  rula 
applies  first?    Rule  5.     What  other  rule  apphes ? 

Result,  du  =  2axy^dx  -f  dax^y^dy. 

2 

Ex.  7.  Differentiate  u  =  6ax^y^. 

1  2. 

Result,  du  =  4,ax~^y^dx  -\-  ISax^y^dy. 
Ex.  8.  Differentiate  y  =  2bz-^  +  Sax'^z'^. 

6 

2  1           'Sax'^dz       Abdz 
Result,  dy  =  5ax^z^dx  + r- • 

^  2^z  ^' 

11  -r.      7     xdy  -f  ydx 

Ex.  9.  Differentiate  u  =  x'^y'^.  Besult,  — ■ — —'- — . 


2^-^i/^ 

P 
Ex.  10.  From  y^  =  2px  find  the  value  of  dy.  dy  =  -dx. 

Ex.  11.  From  A^y-  -\-  B^x^  =  A^B^  find  the  value  of  dy. 

Ex.  12.  From  A^y^  —  B^x^  =  —  A^B^  find  the  value  of  dy. 

dy  =  -— -a^. 

X 

Ex.  13.  From  ^24-2/2  =  R2  find  the  value  of  dy.      dy  = dx. 

Ex.  14.  From  2xy^  —  ay^  £=  x^  find  the  value  of  dy. 

^j,  = ^dx. 

4:Xy  —  "lay 


18  THE  DIFFEKENTIAL   CALCULUS. 

Ex.  15.  Differentiate  u  =  r-~.        Result,  du  = ~ -, 

32/3  Sy^ 

1  dx 

Ex.  16.  Differentiate  v  =  -.  dy  = . 

^        X  x^ 

Ex.  17.  Differentiate  u  =  - — %r—.  du  =        ^^  ^ 


b  —  22/2*  (6  —  2]/2)2* 

Ex.  18.  Differentiate  y  =  —zr--  dy  =  --  x  2xdx  =  -—dx. 

SiTG. — Do  not  treat  this  as  a  fraction  under  Kule  7. 

Ex.  19.  Differentiate  u  ==  x-y^z. 

2.^2  —  3 


Ex.  20.  Differentiate  u  = 


4j7  4-  X' 


Opebation.       dK  =   -^(^^^  -   3)(^-^    +    »^)   -    d(ix    +   x^)(2g_--_3)    _ 

4:Xdx{4:X  +  a;g)  —  (4da;  +  2a;d.-r) (2a;3  —  3)  _  {4x(4x-{-x'^)  —  (4  +  2a;) (2.^^  —  3)} da; 

(8a;^  +  6a;  +  12)da; 
~  (4x  +  a;2)2        * 

Stjg^s. — The  first  step  is  the  application  of  the  rule  for  fractions,  since  the  func- 
tion is  a  fraction  with  a  variable  numerator  and  a  variable  denominator.  The 
second  step  is  to  perform  the  difierentiation  of  2,'r2  —  3,  and  4.x  -|-  s;^.  This  step 
involves  the  rules  for  constant  factors,  variables  affected  with  exponents,  constant 
terms,  and  the  sum  of  variables.  The  remainder  of  the  work  is  reduction  and 
addition  of  terms. 

^     r.^    ^n^        .■  ,                  2a:^                        7          8<22^3  —  4075^ 
Ex.  21.  Differentiate  u  = .  du  =  — —dx. 

(a2  —  x^y 

^       ^                                                  a, 
Ex.  22.  Differentiate  y  = .  dy  = dx. 

/][ 2j7 x'^^dx 

Ex.  23.  Differentiate  y  =  ^   '  '^  .  dy  == rr-— r 

^         t     <    -">  ^  (1  _|_  a;2) 

Ex.  24.  Differentiate  y  = 

Ex.  25.  Differentiate  v  =  . 

^         1  —  X 

Ex.  26.  Differentiate  y  ==:  Sx"^  —  4.  dy  ==  Smx'^-'dx. 

m  29712    ">—" 

Ex.  27.  Differentiate  y  =  2mx\  dy  == x  «  dx. 

^  n 


a-'  — 

-  x^' 

a  — 

X 

X 

1  + 

X 

1  + 

x^ 

1  + 

372 

1 

^2* 

X 

DIFFERENTIATION  OP  ALGEBRAIC  FUNCTIONS.  19 


m    1 


Ex.  28.  Differentiate  w  =  Sno;"!/". 

m — n    1  v%    1 — n 

du  =  1mx~^y^doc  +  2x''y~dy. 

1  71 

Ex.  29.  Differentiate  i/  =  — .  dy  = zTl^^' 

Ex.  30.  Differentiate  y  =  ^ x:^  —  a^. 

Opebation.  — By  the   special  rule  for  the  square  root   (57),   we  have  dy  == 
d(iK3  —  a:^  3x-cZx 


Ex.  31.  Differentiate  y  =  V~ax  ■\-  Vc^. 

ad.v        3c-.r-dr       , ,   i  -i      3c  -i-,  ,          a^  -f  3ca7 
^2/  =  — -^  +  —7=  =  {W^      +  17^  )^^>  o^ -=-dx. 

iVax  ■   Wc:^x^  ^  2va7 

/-        X 
Ex.  32.  Differentiate  y  =  av  a;  —  -. 

o 


Ex.  33.  Differentiate  y  =  V ax  -\-  hx'^  -\-  cx^. 


Ex.  34.  Differentiate  y  =  {ax-  —  x^y. 

Solution. — Kegarding' «x2  —  x^  as  a  variable,  it  is  affected  with  the  exponent  4  ; 
hence  we  have  dy  =  ^{ax'^  —  x^)'^  X  d^ax-  —  x^),  the  operation  of  differentiating 
the  variable  ax-  —  x^  being  as  yet  unperformed.     Performing  this  operation  and 
reducing,  we  have  dy  =  4(aa;-  —  x^Y  X  (2ax  —  ^x'^)dx  = 
8ax^(a  —  xydx  —  12a;8(a  —  xydx. 

Ex.  35.  Differentiate  y=  {a  +  bx")^.        dy==^^-i-{a  f  hx'^)'^bxdx. 

Ex.  36.  Differentiate  y  ==  (a^  +  ^2)3.  dy  =  6x{a''-  +  x'^ydx. 

a  ,  6a^       - 

Ex.  37.  Differentiate  y  = r-  dy  =  —  - — ■ — -r-dx. 

•^        {ly^  +  x^y^  {b-  +  ^2)'« 

Ex.  38.  Differentiate  7/  =  (1  +  2j72)(1  -}-  4^;^). 

Solution.— Regarding  this  function  as  the  product  of  the  two  variables  1  -|-  2x^ 
and  1  +  4x^  we  have  dy  =  d^l  +  2x^)  X  (1  +  4.x^)  +  d(l  +  4.^3)  X  (1  +  Sx^). 
Performing  the  operation  of  differentiating  1  +  2.1?^  and  1  +  4.x^,  we  have  dy  = 
4a;(l  4-  4a;3)(2a;  +  12x2(1  -f  2£c2)da;  =  4a;(l  +  Zx  +  10x3)dx. 

Ex.  39.  Differentiate  y  =  {x^  -\-  a.){Sx^  +  6). 

dy  =  (1507^  +  35^2  +  6ax)dx. 

x^                            ,         3^2  4-  ^3 
Ex.  40.  Differentiate  y  =  7-— -.  dy  =  — — -^^. 

^         (1  +  :r)2  ^         (1  +  ^)3 

a  ,  3a^.r 

Ex.  41.  Differentiate  y  ==  -t r-.  dy 


(a  —  ar)3  (a  —  :r)^ 


20                                     THE   DLFFEBENTIAL   CALCULUS. 
Ex.  42.  Differentiate  y  =  —z -.  dy—  — j^—- — r — • 

^  {ab ^2)3  ^  (a6   —  ^r2)4 

"^x.  43.  Differentiate,  without  first  expanding,  y  =  {1  -^  x)*{l  -\-  x^y. 

dy  =  4(1  +  xy{l  -i-  x^){l  -}-  X  +  2x^)dx. 

Ex.  44.  Differentiate  y  ==  x^  —  \/i  —  x^ 

„    ,  dx'-dx 

av  =  2xdx  -\ ■ 

^  ^  2v/l~^- 

/ : (  n  ^  j  dx 

Ex.  45.  Differentiate  u  ==  v2ax  —  x"^.  du  =  —  - — . 

'^^2ax  —  xi 

Ex.  46.  Differentiate  u  =  Va^  +  ^^  X  Vb"^  +  j/^. 

(^^4-  y^)xdx  +  (a2  +  x-^)ydy 

.      \/a2   _j_   ^-2    X    V   62   +    ?/2 

Ex.  47.  Dinerentiate  i/  =  — -  a^/  = 


v^a2  —  ^2  v/(a2  —  x^y 


Ex.  48.  Differentiate  j/ 


X 


\/l  +  J72 


Sug's.    2/=a;(l+x2)  I    .-.  d?/  =  dx(l +3:2)  ^ -f  a;.  d(l+a;2)  ^==da;(l+a;2)  ^ — 

£c2(l  -j-  a;2)  *dx  =  — ^ =  — -.     Or,  we  may  apply  the  rules 

(1+^2)2  (1+^2)2        

„           -      ,.            ^                           i    XT,        J          dxVl  4-  £c2  —  a;dv/l  +  ic2 
tor  a  fraction    and  a  square    root,  thus    du  =  ■ —. —  = 

xdx 


dXs/l   -f-  iC2  £C 


v^l  +  ^^        dx(l  -f-  a;2)  —  x^dx  dx 


1  _l_  r2  3.  Ji 

(1  — 3^)cZj; 


Ex.  49.  Differentiate  u=  {l-\-x)vl  —  x.         du 


2v'l  — 


X 


Ex.  50.  Differentiate  u  =  —         .  du  = 


\/{i  —  x^y  (1  — ^2)i* 

Ex.  51.  Differentiate  u  =  — . 

2v  a2^2  —  ^4 

—  04(^2  —  2x^)dx 
du= ^. 

2j;2(a2  —  072)2 


Ex.  52.  Differentiate  u  =  \^  x  -\-  \/l  +  x\ 


xdx 


Sug's. — Squaring    u2   =  a;  -}-  s/1  -f-  x'^.      2iLdu   ==  dx  +  — .      du   = 

v/l  +  a:-' 


dx  -j- 


DIFFERENTIATION   OP   ALGEBRAIC  FUNCTIONS.  21 

xdx  (a;   -f   y/l   4-   x')dx 


a;2 
Or,   we  may  differentiate  without   squaring,   thus   du 


2v- 1   4-  a;' 
xdx 

dx  -j y- 

2  Va;  4-  n/IT^^        2 v/r+¥'  V a;  -f  v/r+^'^  ^^^  +  '^^ 


x^dx 


X 

Ex.  63.  Differentiate  u 


V  a'^  -\-  x-^  —  X 

Sug's. — As  the  denominator  is  more  involved  in  the  differential  of  a  fraction 
than  the  numerator,  it  is  expedient  to  reduce  the  fraction  to  a  form  having  as  sim- 
ple a  denominator  as  possible.     Bationalizing  this  denominator,  we  have     u  = 

^2  —  a2  a-^  aVa;-^  +  a^  «^ 

Ex.  54.  Differentiate  w  =       '         . 

V  x'^  4-14-^ 

du  =  2  \  2x g^l±-L  I  dx, 

VX''  4-  1 


11  s/x 

Ex.  55.  Differentiate  u  = 


1  +  \/: 


X 


„     ,  1  —  v/.'T        Vl  —  s/x       s/l  —  X      .  dx 

SUGS.       U  =  K.     z — : =  —  = .       du 


^  +  ^a?        \/l4-v/x       l  +  v^a;  2(l4->/a;)ya;  — a;2 

X     \"  ,  nx''~^dx 


(X       \ 
) .  du  =2 
1  -^  x/ 

Ex.  57.  Differentiate  u  = 


(i  4-  x)"-^'' 


\/l  -i-  X  —  VI  —  X 

1  +  v/l  —  ^^ . 
du  = ; —  dx. 


Ex.  58.  Differentiate  w  =  >/a;  •  V  ^a;  4-  1 


,                  7a;^  4-  4  , 

du  =  — dx. 


22 


THE  DIFFERENTIAL   CALCULUS. 


Ex.  59.  Differentiate  u  =  N/2:r— 1— V25;~l  — \/2a7-^l— -,  etc., 
to  infinity. 


Sug's.— We  liave  u  =  v^'^x  —  1  —  m  ;    whence  m^  =  2x  —  1  ~  m,    and  u 
—  si  iv/8x  —  3.     .  • .  di*  =  ± 


v^Sx  —  3 
Ex.  60.  Differentiate  u  =  ,y 

du 


v/ 


+  V  (c2  —  a;2 


J? 


)']■ 


36 


4^7 


—  "^ 


ILLUSTRATIVE    EliLOIPLES. 

[Note. — The  following  examples  are  designed  to  illustrate  more  fully  the  significance  of  the 
process  of  differentiation.] 

Ex.  1.  In  a  parabola  whose  parameter  is  12,  which  is  increasing 
the  faster  at  a;  =  2,  the  ordinate  or  the  abscissa,  and  how  much  ?  At 
^  =  3  ?  At  ^  =  8  ?  At  ^  =  24  ?  How  does  the  relative  rate  of 
change  vary  as  we  recede  from  the  vertex?  At  what  point  are  ordi- 
nate anid  abscissa  varying  equally  ? 

Solution. — The  equation  of  this  parabola  is  y-  =  12x. 

Differentiating,  -we  have  dy  =  -dx.    Now,  as  differentiat-     ^ 

ing  is  the  process  of  finding  the  difference  between 
two  consecutive  states  of  a  function,  dx  represents  one 
of  the  infinitesimal  increments  of  a;,  as  DD',  and  dy 
the    coniemporaneous,    infinitesimal    increment   of    y,   as 

n 

P'  E.     We,  therefore,  learn  from  dy  =  -dx  that  in  gen- 

eral  dy  is  -  times  as  great  as  dx  ;  or.  in  other  words,  that 
2/ 

a  — 

y  changes  _  times  as  fast  as  x.     At  P  where  a;  =  2,  y  ■=  \/24.     Hence,  at  this 

y     6 
point,  dy  =  — ■z=dx  =  ^v/edx  ;  that  is,  y  is  increasing  nearly  li  times  as  fast  as  x. 

>/2i 
At  P"  where  re  =  3,  ?/  =  6,  and  dy  =  dx  ;  that  is,  x  and  y  are  increasing  equally. 
In  general,  at  the  focus  the  ordinate  and  abscissa  of  a  parabola  are  increasing 
equally,  since  at  this  point  y  =^  p.  At  P''^  where  cc  =  8,  y  is,  increasing  only  about 
.G  as  fast  as  x.  At  P^'  where  cc  =  24,  y  is  increasing  at  the  still  slower  rate  of 
about  .35  as  fast  as  x.     Finally,  it  is  evident,  fi'om  a  slight  inspection  of  the  figure, 


Fig.  11. 


DIFFERENTIATION   OF  ALGEBRAIC   FUNCTIONS.  23 

that  y  increases  less  and  less  rapidly  as  x  becomes  larger,  x  continuing  to  increase 
at  a  uniform  rate.  At  cc  =  co,  y  ceases  to  increase,  i.  e.  the  branches  become  par- 
allel to  the  axis  of  x.- 

Ex.  2.  Examine  the  relative  rates  of   change  of  the  ordinate  and 
abscissa  in  the  eUipse. 

JB-x 

Solution.— Differentiating   A-y^  +  B-x"^  =  A'^B'^,  we  find  dy  =  —  -jr-dx  — 

-A-  y 
p., 

:dx.     On  this  we  observe  1st,  That  the  —  sign  shows  that  x  and  y 


AVA^  —  a;2 

are  decreasing  functions  of  each  other  ;  that  is,  that  as  a;  takes  an  increment  y 

takes  a  decrement.     This  is  evident  from  a  consideration  of  the  curve.     2nd,  That 

Bx 
in  general  terms  y  diminishes times  as  fast  as  x  increases.     3rd,  At 

A\^A^  ~  a;2 

X  =  0,  I  e.  at  the  extremity  of  the  conjugate  axis,  y  is  not  increasing  or  decreas- 

Bx 
ing,  since  here =  0,  and  dy  =  Q  -dx  —  O.     At  the  extremity  of  the 

As/A^  —  x^ 

transverse  axis  dy  =  —  oc  •  dx,  i.  e.  y  is  decreasing  infinitely  faster  than  x  increases. 

There  are,  therefore,  all  relative  rates  of  change  between  x  and  y  from  0  to  oc. 

Moreover  as  x  begins  to  increase  from  0,  y  commences  to  decrease  (at  first  slowly, 

Bx 
as  the  fraction  • — is  small  when  x  is  small),  and  then  more  and  more 

AVA^  —  x'^ 
rapidly  as  x  increases,  till  it  reaches  an  infinitely  rapid  rate  of  decrease  at  x  =  A. 

Bx 
This,  it  is  easy  to  see,  is  the  law  of  change  in  the  fraction  —      _    as  x  in- 

AvA'^  —  x'^ 

creases.  The  same  law  is  also  rendered  probable  from  an  inspection  of  the  curve. 
Finally,  we  may  inquire  at  what  point  the  relative  rates  of  change  sustain  any 
given  relation  to  each  other,  as,  for  example,  when  y  decreases  twice  as  fast  as  x  in- 
creases, or  just  as  fast,  or  10  times  as  fast.     Thus  when  y  decreases  twice  as  fast  as 

Bx 
X  increases,  we  must  have  dy=  —  Mx,  i.  e. ^rr  =  2.     From  this  we  find 

2A^ 

a;  =  i  . —  .   -  ;    hence  at  these  points,  y  is  diminishing  twice  as  fast  as  x  is 

v/4^2  -f  J52 
increasing. 

Ex.  3.  A  boy  is  running  on  a  horizontal  plane  directly  toward  the 
foot  of  a  tower  100  feet  in  height.  How  much  faster  is  he  nearing  the 
"foot  than  the  top  of  the  tower?  How  far  is  he  from  the  foot  of  the 
tower  when  he  is  approaching  the  base  twice  as  fast  as  he  approaches 
the  top  ?  How  far  off  must  he  be  to  be  approaching  both  base  and 
top  equally  ?  Where  is  he  when  he  is  not  approaching  the  top  at  all, 
or  is  making  infinitely  more  progress  toward  the  base  than  towards 
the  top?  When  he  is  at  200  feet  from  the  base  of  the  tower  how 
much  faster  is  he  approaching  the  base  than  the  top  ? 

Sug's. — Let  AB  represent  the  tower,  and  AX  the  line  in  the  plane  of  the  base 
in  which  the  boy  is  approaching  the  base.     Suppose  the  boy  at  any  point,  as  P 


24  THE   DIITEEENTLIL   CALCULUS. 

and  let  AP  =^  x,  and  PB  =  y.      Then 

2/2  — .-r^rrr  10000.    WhencB  dy  =  -dx.    Hence 

we  see  that  in  general  he  is  only  approach- 

ing  the  top  an  -th  part  as  fast  as  he  is  the 

base  ;  i.  e.,  letting  PP'  represent  an  infin-  ^^'^^  ^^' 

itesimal  element  of  the  distance  to  the  foot  of  the  tower,  P  F  represents  a  contem- 
poraneous, infinitesimal  element  of  the  distance  to  the  top  ;  and  also,  that  P  F    is 

X 

an  -th  part  of  PP'.     Secondly,  when  he  is  approaching  the  foot  of  the  tower 

1  X       1 

twice  as  fast  as  he  is  the  top  ;  we  have  dy  =  ^dx,  or  -  =  -  ;  whence  y  =  2cc.     But 

^  2  2/2 

100 
2/-  —  x^=^  10000  ;  and,  substituting,  3a;'-  =  10000,  or  x  =  — -  =  uS  nearly.     Lastly, 

\/3 

/J.       200       25 

when  he  is  at  200  feet  from  the  base  y  =  \/50000  =  224  nearly,  and  -  =  -—  =  — 

y       224       2o 

25  25 

nearly.     Hence  dy  =  ^-^^x,  or  he  is  approaching  the  top  —  as  fast  as  he  is  the 
28  2o 

base.     [Let  the  pupil  decide  the  other  points  himself.  ] 

Ex.  4.  A  sliip  is  sailing  northwest  at  15  miles  an  hour.     At  what 
rate  is  she  making  north  latitude  ? 

An^.,  At  10.6054-  niiles  an  hour. 

Sitg's. — Let  2/  represent  any  distance  run  in  the  northwest  course,  and  x  the  cor- 
responding northing.     Then  as  the  course  is  northwest  there  is  made  in  the  same 

time  X  westing,  and  we  have  y^  =  2x-.     From  this  dy  =  — dx,  and  the  ship  is  run- 

2x  —  2x 

3iing  --  times  as  fast  as  she  is  making  northing.     But  y  =  x\/%  whence  -—  = 

^x  -  -  - 

— ^:  =  \/2,  and  dy  =  \^1dx,  or  dx  =  ^V'ldy ;   i.  e.,  she  is  making  northing 

a:\/'2 

,  707-f-  as  fast  as  she  is  running. 

Ex.  5.  In  the  function  y  =  27x  -|-  8^72,  required  the  value  of  x  when 
y  is  increasing  45  times  as  fast  as  :3t.  Result,  x  =  o. 

Ex.  6.  What  is  the  relative  rate  of  variation  of  the  side  and  alti- 
tude of  an  equilateral  triangle?  i.  e.,  if  the  side  takes  an  infinitesi- 
mal increment,  what  is  the  contemporaneous  infinitesimal  increment 
of  the  altitude  ?  When  the  side  is  increasing  at  the  rate  of  2  inches 
per  second  how  rapidly  is  the  altitude  increasing  ?  Is  the  relative  rate 
of  increase  constant  or  variable  ;  that  is,  does  the  altitude  increase 
more  or  less  rapidly  in  comparison  with  the  side  when  the  side  is 
small  than  it  does  when  it  is  large,  or  is  the  relative  rate  of  increase 
always  the  same? 

Sug's.  —Let  y  =  the  altitude  and  x  one  cf  the  sides  of  the  tdangle.     Then 


LOGARITHMIC  AND  EXPONENTIAL  FUNCTIONS.  25 

3x  Sx  ~ 

?/2  =  |aj2  and  di/  =  ~-dx  = ^-dx  =  i\/ddx.     Hence  we  see  that  the  infinitesi- 

4?/  2s/ '3x 

mal  increment  of  y  is  always  i\/3  times  as  much  as  the  contemporaneous  infini- 
tesimal increment  of  x.  When  x  is  increasing  at  2  inches  per  second  y  is  increas- 
ing 2>/3  times  2  inches,  or  \/3  inches  per  second. 

ScH.  — The  student  should  now  be  able  to  comprehend  with  considerable 
clearness  the  object  of  the  Differential  Calculus;  viz.,  having  given  the 
relation  between  finite  values  of  variables,  to  find  the  relation  between  the 
contemporaneous  infinitesimal  increments  of  those  variables,  or  their  rela- 
tive rate  of  cl^ange.  Thus,  in  the  last  example,  the  relation  between  the 
altitude  and  one  side  of  an  equilateral  triangle,  3/2  =  lx~,  is  the  relation 
between  finite  values  of  the  variables,  from  which  we  find  the  relation  be- 
tween the  contemporaneous  infinitesimal  increments  d^  and  dx,  by  the  Dif- 
ferential Calculus. 


■4»»- 


SUCTIOJSr  IL 
Differentiation  of  Logaritlmlio  and  Exponential  Functions.* 

S8,  The  JI£odulus  of  a  system  of  logarithms  is  a  constant 
factor  which  depends  upon  the  base  of  the  system  and  characterizes 
the  system. 

SO,  I^vop. — The  differential  of  the  logarithm  of  a  variable  is  the 
differential  of  the  variable  multiplied  by  the  modulus  of  the  system,  di- 
vided by  the  variable  ;  or,  in  the  Napierian  system  the  modulus  being  1, 
the  differential  of  the  logarithm  is  the  differential  of  the  variable  divided 
by  the  variable. 

Dem. — Let  2/ =  a*.",  n  being  constant.      Then  log  y  =  n  log  a;.     Differentiating 

dy 

y  =  x",  we  have  dy  =  nx^—^dx,  orn  = fr-  =  — -  =  -r->  since  X"—^  =  --.     Again, 

^  .^  x'^-Hx      y  ^        dx  X        ^ 

-dx      — 

X  X 

whatever  the  differentials  of  log  y  and  log  x  are,  we  have  d{log  y)  ■=  n  -  dQiOgx), 

ox   n  =  -\         .      Placing  these  values  of   n  equal  to  each  other,   we  obtain 

d(logic)  °  \ 

dy 

-  ,  '     .  =  -— .     Now  let  m  be  the  factor  by  which  --  must  be  multiplied  to  make 
a(logfl;)       dx  V 

X 

it  equal  to  ^(log?/),  then  is  (^(logic)  = . 

*  See  3^,  33, 


26  THE  DIFFERENTIAL  CALCULUS. 

"We  are  now  to  show  that  m  is  a  constant  depending  upon  the  base  of  the  system. 

To  do  this  take  y  ^z""',  from  which  we  find  as  before  n  =   /  °^^^  =  ~.     But  m 

a{log  z)       dz 

z 
is  the  ratio  of  d{logy)  to  —  ;  hence  d(logz)  = .     Thus  we  see  that  in  any  case 

y  2 

*the  same  ratio  exists  between  the  differential  of  the  log.  of  a  number,  and  the  differ- 
ential o!  the  number  divid^-d  by  the  number.  Therefore  m  is  a  constant  factor. 
To  show  that  m  depends  il^)wu  tue  base  of  the  system  we  have  but  to  recur  to  the 
definition  of  a  logaritlim  to  see  that  the  only  quantities  involved  are  ihe  number,  its 
logarithm,  and  the  base  of  the  system.  Of  these  the  two  former  are  variable,  whence, 
as  the  base  is  the  only  constant  in  the  scheme,  m  is  a  function  of  the  base.  * 

Finally,  as  m  depends  upon  the  base  of  the  system,  the  base  may  be  so  taken 
that  m  :=  1.  The  system  of  logarithms  founded  on  this  base  is  called  the  Napie- 
rian system,     q.  e.  d. 


00,  JPtoj)- — The  differential  of  an  exponential  function  with  a 
constant  base  is  the  function  itself,  into  the  logarithm  of  the  hose,  into  the 
differential  of  the  exponent,  divided  by  the  modulus. 

Dem. — Let  2/ =  «'.     Taking  the  logarithms  of  both  members  log  2/ =  £c  log  a. 

Differentiating  — -  =  log  adx,  or  dy  = ^^ ,  remembering  that  y  =  a^,  and 

that  log  a  is  constant,     q.  e.  d. 

Ql,  Cor.  1. — The  differential  of  an  expjonentiol  function  with  a  con- 
stant base,  taken  with  reference  to  the  Napierian  system,  is  the  function 
itsef,  into  the  logarithm  of  the  base,  into  the  differential  of  the  exponent. 
Thus  if  J  ==  a"",  cly  =  p/  log  adx. 

02 »  Cor.  2. — If  the  base  of  the  exponential  is  the  base  of  the  system  cf 
logarithms  in  reference  to  ivhich  the  differeyitiation  is  made,  we  have,  in 

general,  dy  = ,  or  in  the  Najjierian  system  dy  =  e'^dx,  since  the  log- 
aritlim (f  the  base  of  a  system,  taken  in  that  system,  is  1,  and  in  the  Na- 
pierian system  e  is  used  to  represent  the  base  and  m  =  1. 


G3,  JPvo2>o — The  differential  of  an  exponential  with  a  variable  base 
is  best  obtained  by  passing  to  logarithms,  and  then  differentiating. 

III.. — Let  w  =  2/"".     Passing  to  logarithms,  log  it  =  a;  log  y.     Differentiating,  we 

,  .  ,   mdu        ,  ,      ,    mxdy      ,  ,  ulogy  dx    ,    u  x  dy 

have  m  neneral, =  lo!?  y  dx  4- ,  whence  du  =  +  —  = 

-^  u  y  m  '        y 

11^  loc  11  dx        ii^xdii 

' ^^-^ — '■ 1-  — — '-.     If  the  logarithms  are  taken  in  the  Napierian  system,  m  =  1, 


*  What  tliis  relation  is,  it  does  not  concern  us  at  present  to  know.    It  will  be  determined  here- 
after. 


LOGARITHMIC  AND  EXPONENTIAL  FUNCTIONS.         27 

and  du  =  y*  log  y  dx  -{-  y—^xdy.     If  in  addition  y  =  x,  so  that  u  =  a;*,  du  == 
x*(loga;  -f-  l)dx. 


EXERCISES. 

[Note. — The  following  exercises  are  designed  to  familiarize  the  rules  for  differentiating  loga- 
rithmic and  exponential  functions,  and  give  the  needed  facility  in  applying  them.] 

Ex.  1.   Differentiate  u  =  x  log  x. 

du  =  logxdx  -\-  mdx,  or  {logx  +  l)dx. 

Ex.  2.  Differentiate  u  ==  log  x^  du  =  2m--,  or . 

°  XX 

d X  dx 

Ex.  3.  Differentiate  u  =  log^  x.       du  =  2m  log  x- — ,  or  2  log  x — . 

X  X 

Ex.  4  Differentiate  u  =  x""'. 


du  =  af'x'  ]  log  x{\og  ^  4-  1)  +  -  r  dx/ 


a}°^  -^  loff  a  , 
Ex.  5.   Differentiate  u  =  a}"^',  du  == ^-dx. 

X 

, xdx 

Ex.  6.  Differentiate  u  =  losf  \/l  —  x^.  du  =  — . 

^  1  —  x^ 

# 

6^+1-7 
Ex.  7.  Differentiate  u  =  log  (3x^  +  x).  du  = ■ dx. 

^  ^  ^  6x^  +  X 

Ex.  8.  Differentiate  u  =  log  {x  -\-  \/l  +  x'^).         du  = 


yi  +a;2 


2 
Ex.  9.  What  is  the  differential  of  w  =  a""  in  the  common  system 

2    2 
when  a  is  the  base  of  the  system  ?  du  =  —  a""  xdx. 


m 


Ex.  10.  Differentiate  u  =  e^"^"",  in  the  common  system,  e  being  the 

.     ,                                                     udx .  mudx 

base  of  the  Napierian  system.  du  = log  e  ==  — — . 

SuG. — If  the  student  has  studied  the  subject  of  logarithms  as  usually  presented 
in  our  higher  Algebras,  he  has  learned  that  the  common  logarithm  of  the  Napier- 
ian base  is  the  modulus  of  the  common  system  ;  L  e. ,  in  this  example  log  e  =  m. 
This  fact  will  also  appear  hereafter. 


V  X'i  -\-    I    X 

Ex.  11.  Differentiate  u  =  log • 

V  X-  4-  1  +  ^ 
SuG.— First     rationalize    the    denominator    of   the    fraction,    obtaining    u 

Mx 


log  (\/x-2 4- 1  —  ic)2  =  2  log( \/x2 -)- 1  —  x),  and  then  differentiate,     du  =  — 


\/iC2-j-l 


*  The  student  will  observe  for  himself  whether  common  or  Napierian  logarithms  axe  used. 


28  THE  DIFFERENTIAL  CALCULUS. 

64»  ScH. — The  differentiation  of  algebraic  functions  is  often  performed 
with  greater  faciUty  by  first  passing  to  logarithms. 

Ex.  12.  Differentiate  u  = . 

1 — x^ 

SuG.  — Passing  to  logarithms  we  have  log  u  =  log  (1  -f-  x^)  —  log  (1  —  ^-)-     Biffer- 

du        2xclx       —  2xdx                 4:Xdx  Axdx 

entiatmg,    —  =  — - — =  —— — — tj —-     .' .  clu 


X  — ^^^-  = — ^ — .     This  example  illustrates  the  method 


(1  -\-x-^){l  —  x-^)       1  —  x^       (1  —  rC2)2 
referred  to  in  the  scholium,  although  the  student  will  find  the  direct  method  quite 
as  expeditious. 


Ex.  13.  Differentiate  u  =  x{a^  +  x^)va^  —  x%  by  first  passing  to 

logarithms.  du  = —  — dx. 

va^  —  X- 

Ex.  14.  Differentiate  w  =  (a"  +  l')^.        du  =  2a"(a"  +  1)  logadx. 

a""  —  1  -  ,  2^""  log-  adx 

Ex.  15.  Differentiate  u  ==  —- -.  du  =  -. 

a''  -^  1  {a''  +  1)2 

66.  Cor. — The   ordinary  rule    (SO)  for   differentiating  a  variable 
affected  with  an  exponent  applies  when  the  exponent  is  imaginary. 

Dem  — Let  u  =  x^  *'— ^    Passing  to  logarithms,  log  u  =  a  V —  1  log  x.     Differentiat- 

du            , — -dx             ,              . — -xidx  , — -   aVZl_i 

mg,  —  =  av  —  1 — .     .  • .  dit  =  av  —  1 =  av  —  1  x  dx.     q.  e.  d. 


ILLUSTRATIYE   EXAMPLES. 

Ex.  1.  Which  increases  the  faster,  a  number  or  its  logarithm  ? 

Solution.  — Let  x  represent  any  number  and  y  its  logarithm,  so  that  y  =  log  x. 
"We  now  wish  to  find  the  relation  between  the  contemporaneous,  infinitesimal  in- 
crements of  X  and  y  ;  i.  e.,  if  the  number  {x)  changes  how  does  the  logarithm  (y) 

change?     Hence  we  differentiate,  and  have  dy  =  -dx.     Prom  this  we  see  that  the 

increment  of  the  logarithm  {dy)  is  —  times  the  increment  of  the  number  {dx). 

Therefore  when  ic  <<  m  the  logarithm  increases  faster  than  the  number ;  when 
X  >  m  t'le  logarithm  increases  more  slowly  than  the  number  ;  and  when  x  =  m 

they  incujabe  equally. 

[Note. —The  student  should  not  fail  to  see  in  every  such  example  the  real  object  of  the  Dif- 
ferential Calculus  {42).  In  the  last  example  the  relation  between  finite  values  of  the  A-ariables  x 
and  v  is  y  =  log.  x.  The  relation  between  the  contemporaneous,  infinitesimal  elements  of  these 
variables  is  found  by  differentiating,  this  being  the  object  of  the  Differential  Calculus.] 

Ex.  2.  When  the  number  is  2124  and  is  conceived  as  passing  on  to 
larger  values  by  the  law  of  growth  of  continuous  number,   i  e.  by 


LOGARITHMIC  AND  EXPONENTIAL  FUNCTIONS.  29 

taking  on  infinitesimal  increments,  liow  much  faster  is  the  number 
increasing  than  its  common  logarithm  ?  If  this  relative  rate  of  change 
continued  uniform  (which  it  does  not)  while  the  number  passed  to 
2125,  i.  e.  increased  by  1,  how  much  would  the  logarithm  have  in- 
creased ? 

Solution. — Letting  x  be  any  number  and  y  its  logarithm,  we  have  found  that 

dy  =  —dx.     But  m,  the  moduhis  of  the  common  system  =  .434:29448.     Hencef 

X 

. 43429448 
when  X  =  2324,  we  have  dy  =  '- — — — — dx  =  .000204dc,  or  the  increment  of  the 

logarithm  is  .000204  part  of  the  increment  of  the  number.     The  number  is,  there- 

1000000 
fore  increasing  — — — — ,  or  about  4902  times  as  fast  as  the  logarithm.     Secondly, 

If  this  relative  rate  of  change  continued  the  same  while  the  number  passed  from 
2124  to  2125,  I.  e.  increased  by  1,  the  logarithm 
would     increase     once    .000204,     or    .000204. 
Hence  the  logarithm  of  2125  would  be  .000204 
larger  than  the  logarithm  of  2124. 

GEOMETEicAii   Illustbation. — Let    M  N   be 
the  curve  whose  equation  is  y  =  log  x.     Take 
A  D  =  2124  ;  then  will  P  D  represent  its  loga- 
rithm.     Let    D  D '   represent    dx ;    then  will  p       -.  o 
P'E  represent  (Z?/.* 

Ex.  3.  The  common  logarithm  of  327  is  2.514548.  What  is  the  log- 
arithm of  327.12,  on  the  hypothesis  that  the  relative  rate  of  change 
of  the  number  and  its  logarithm  continues  uniformly  the  same  from 
327  to  327.12  that  it  is  at  327? 

43429448 
SuG.— At  327  dy  = '- — — — dx  =  .001328d'r.     Now  as  the  number  327  increases 

.12  to  become  327.12  ;  according  to  the  hypothesis  the  logarithm  increases  .12 
times  .001328  or  .000159.     Hence  the  logarithm  of  327.12  is  2.514707. 

ScH. — The  hypothesis  that  the  relative  rate  of  change  of  a  number  and 
its  logarithm  continues  constant  for  comparatively  small  changes  in  the  num- 
ber, is  sufficiently  accurate  for  practical  purposes,  and  is  the  assumption  made 
in  using  the  tabular  difference  in  the  table  of  logarithms,  as  explained  in  The 
Complete  School  Algebka  {125),  and  in  the  introduction  to  the  table  of 
logarithms  {14:)  in  the  volume  on  Geometry  and  Trigonometry. 

Ex.  4.  What  should  be  the  tabular  difference  in  the  table  of  loga- 
rithms for  numbers  between  2688  and  2689  ?         Ans.,  .00016156+. 

QuEKT. — How  is  it  that  the  tabular  difference  found  in  the  table  of  logarithms  for 

*  The  figure  is  necessarily  out  of  proportion,  as  the  true  relation  of  y  and  x  requires  that  A  D 
be  nearly  700  times  as  long  as  PD. 


30  THE  DIFFERENTIAL  CALCULUS. 

numbers  between  2688  and  2689,  is  162?  Sbow  how  the  method  of  nsing  this 
tabular  difference  makes  the  result  agree  substantially  with  the  method  of  inter- 
polating now  being  presented. 

Ex.  5.  According  to  the  arrangement  of  our  common  tables,  show 
that  the  tabular  difference  corresponding  to  7487  is  58. 


■#♦» 


SUCTION  III 
Differentiation  of  Trigonometrical  and  Circular  Functions. 

TEIGONOMETKICAL    FUNCTIONS. 

SG,  J^TOp, —  The  differential  of  the  sine  of  an  arc  {or  angle)  is  the 
cosine  of  the  same  arc  into  the  differential  of  the  arc. 

Dem. — Let  X  be  any  arc  (or  angle)  and  y  its  sine,  i.  e.  let  y  =  sin.r.  If  a;  takes 
an  infinitesimal  increment  (dx),  let  dy  represent  the  contemporaneous  infinitesi- 
mal increment  of  y.  Then  the  consecutive  state  of  the  function  is 
y  -\-  dy  ^  sin  {x  -\-  dx)  =  sin  x  cos  dx  -j-  sin  dx  cos  x. 
Now  cos  da;  =  1,  since  as  an  angle  grows  less  its  cosine  approaches  the  radius  in 
value,  and  at  the  limit,  is  radius.  Moreover,  as  an  angle  grows  less  the  sine  and  the 
corresponding  arc  approach  equality,  and  at  the  limit  we  have  sin  dx  =  dx.*  The 
consecutive  state  may  therefore  be  written  y  -{-  dy  =  sin  x  -|-  cos  x  dx. 

From  this  subtract  y  =  sin  x 

and  we  have  dy  =  cos  x  dx, 

which,  being  the  difference  between  two  consecutive  states  of  the  function,  is  the 
differential,     q.  e.  d. 


67 »  JPvop, — The  differential  of  the  cosine  of  an  arc  (or  angle)  has 
the  opposite  algebraic  sign  from  the  function,  and  is  numerically  equal  to 
the  sine  of  the  same  arc  into  the  differential  of  the  arc. 

Dem. — Let  x  be  any  arc  (or  angle)  and  y  its  cosine,  so  that  y  =  cos  a.  Since 
cos  X  =  sin  (90°  —  x)  we  have  y  =  sin  (90°  —  x).  Differentiating  this  by  the  pre- 
ceding proposition,  we  obtain 

dy  =  cos  (90°  —  cc)  X  d{90°  —  x)  =  cos  (90°  —  x){—  dx)  =  —  sin  ic  dx, 
since  cZ  90°  —  x)  =  —  dx,  and  cos  (90°  —  x)  =  sin  x.     q.  e.  d. 

68,  ScH. — The  opposition  in  the  signs  of  the  differential  of  the  cosine, 
and  of  the  corresponding  arc,  signifies  that  they  are  decreasing  functions 
of  each  other  [40)  ;  ^.  e.,  if  one  takes  an  incremeyit  the  other  suffers  p  de- 
crement. 

*  The  student  m&Y  be  inclined  to  say  that  at  the  limit  siu  dx  =  0.  This  is  time,  and  no  error 
would  follow  from  the  assumption  ;  but  the  statement  in  the  demonstration  is  equally  true,  and 
we  consider  sin  dx  =  dx  instead  of  =  0,  simply  because  we  do  not  wish  to  have  dx  vanish  from 
the  formula,  our  object  being  to  find  the  relation  between  dy  and  dx. 


TRIGONOMETEICAL   FUNCTIONS.  31 

69 •  JProp* — The  differential  of  the  tangent  of  an  arc  (or  angle)  is 
the  square  of  the  secant  of  the  same  arc  into  the  dfferential  of  the  arc ; 
or  for  the  square  of  the  secant  we  may  write  the  reciprocal  of  the  square 
of  the  cosine. 

Dem. — Let  11  =  tana;.     Now  tan  x  = '-,  whence  y  = '-.     Differentiatine 

■^  cos  X  cos  X 

Sin  ^ 

this,  observine  that '-  is  a  fraction  with  a  variable  numerator  and  denominator, 

cos  a; 

and  hence  can  be  differentiated  by  the  rule  for  fractions  {54:),  and  the  two  propo- 

^^    ^^.         -,          ,         cos  cede  sin  a;') — sin.'rd(cosa;)       coH-xdx-\-s\u'^xdx 
sitions  (oo.  07),  we  have  ay  = = ■ 

■^  ^  cos2a;  cos- a; 

cos2  X  4-  sin2  .r  ,  1      , 

= ■ ax  = ax  =  sec'^xax.     q.  e.  d. 

cos2  a;  cos^  a; 

Another  Demonstbation.  — The   consecutive  state  of  the  function  y  =  tan  x, 

,    .  .     ,  ,  ,     -,  V  tan  =r  4- tan  di3  isnax  4- d.v      ,,      _,„ 

being  y  -\-  dy  =  tan  (a;  +  dx)  = ; , = -,->  the  difference 

1  —  tan  ic  tan  da;         1  —  tan  a;  aa; 

between  the  two  states,   i.  e.   the  differential  is  dy  = -^ — ^ tan  x  = 

1  —  tanxaa; 

^  _j_  tan-  X 

—dx  =(14-  tan2a;)di;  =  sec~xdx.     [Let  the  student  give  the  detailed 

1  —  tan  xdx  . 

explanation  of  the  process.  ] 


70.  JPvop, — The  differential  of  the  cotangent  of  an  arc  (or  angle) 
has  the  opposite  algebraic  sign  from  the  function,  and  is  numerically  equal 
to  the  square  ofjhe  cosecant  of  the  same  arc  into  the  differential  of  the 
arc  :  or  for  the  squafe  cf  the  cosecant  we  may  turite  the  reciprocal  of  the 
square  of  the  sine. 

Dem. — Let  y  =  cot  x  =  tan  (90°  —  a;).     Differentiating  by  the  last  proposition 
dy  =  sec2  (90^  —  x)  X  cZ(90o  —  a;)  =  cosec^  x{ —  dx)  =  —  cosec^  a;da;,  or r—^dx. 

Q.  E.  D. 

cos  X 


[Let  the  student  demonstrate  this  rule  by  remembering  that  y  =  cot  x  = 


Binx 

and  also  by  taking  the  difference  between  the  consecutive  states  y  =  cot  x,  and 
y  ~\-  dy  =  cot  (x  -{-  dx),  developing  and  reducing  as  in  the  second  demonstration 
under  {69).} 

Query. — What  is  the  significance  of  the  opposition  in  signs? 


71»  JPvop, — The  differential  of  the  secant  of  an  arc  (or  angle)  is 
the  tangent  of  the  same  arc  into  its  secant  into  the  differential  of  the  arc. 

Dem. — Let  y  =  sec  a;  =  .     Differentiating  by  {55,  67)  "we  have  dy  = 

sin  a;  da;       sin  a;  1  ,         , 

= X  X  dx  =  tan  x  sec  a;aa;.     q.  e.  d. 

cos2  X        cos  x       cos  X 


82  THE  DIFFERENTIAL  CALCULUS. 

72*  JPvop. —  The  differential  of  the  cosecant  of  an  arc  (or  ang>)  ],as 
the  opposite  algebraic  sign  from  the  function,  and  is  numerically  equal  to 
the  cotangent  of  the  same  arc  into  its  cosecant  into  the  di^erential  of  the  arc. 

Dem. — Let  y  =  coseccc  =  sec  (90°  —  x).     Differentiating  by  the  last  proposition, 
dy  =z  tan(90°  —  a;)sec(90°  —  x)d{'dO°  —  a;)  =  cot£ccosecjK( — dx)  =^  —  cotxcosecadx. 

Q.  E.  D. 

[Let  the  student  demonstrate  this  proposition  from  the  relation  y  ^  cosec  x  = 

sin  x' 

Query. — What  is  the  significance  of  the  opposition  in  signs? 


73,  IPTOp,—  The  differential  of  the  versed-sine  of  an  arc  (or  angle) 
is  the  sine  of  the  same  arc  into  the  differential  of  the  arc. 

Dem.— Let  y  =  versic  =1  —  cosic.    Differentiating  by  {S7),  dy  =  smxdx. 

Q.   E.   D, 

QuEEY.  — Why  should  the  differential  of  the  versed-sine  be  numerically  the  same 
as  the  differential  of  the  cosine,  but  have  an  opposite  sign  ?  Illustrate  geometri- 
cally. 


74.  J^vop, — The  differential  of  the  coversed-sine  of  an  arc  {or  angle) 
has  the  opposite  algebraic  sign  from  the  function,  and  is  numerically  equal 
to  the  cosine  of  the  saine  arc  into  the  differential  of  the  arc. 

Dem. — (Similar  to  the  preceding.) 

QuEBY. — Why  should  the  coversed-sine  have  the  same  differential  as  the  sine, 
but  with  an  opposite  algebraic  sign  ?     Illustrate  geometrically. 


EXERCISES. 

Ex.  1.  Differentiate  u  =  sin  x  cos  x. 

SuG. — Observe  that  we  have  here  the  product  of  two  variables,  viz.,  since  and 
cos  X.  Hence  du  =  cos  .-c  d(sin  x)  -f  sin  x  d(cos  x)  =  cos-  xdx  —  sin^  xdx  = 
(cos^a  —  sin2  x)dx,  or  (2  cos2  x  —  l)dx,  or  (1  —  2  sin^  x)dx,  or  cos  2  x  dx. 

•  Ex.  2.  Differentiate  u  =  cos^a;. 

SuG. — Observe  that  this  is  the  cube  of  the  variable  cos  re.  Hence  apply  (56) 
and  we  have  dii  =  3  cos^  x  d{cos  x)  =  —  3  cos^  re  sin  a  dx  =  3(sin3  x  —  sin  x)dx. 

Ex.  3.  Differentiate  u  =  tan  5x. 

SuG.     du  =  sec2  5xd{5x)  =  5  sec2  5x  dx. 

Ex.  4.  Differentiate  u  =  cot^  x^-.       du  =  —  Qx^  cot  x^  cosec^  x^  dx. 

Ex.  5.  Differentiate  w  ==  sins  x  cos  x. 

du  =  sin2  x(^  —  4  sin^  x)dx. 


TEIGONOMETEICAL  FUNCTIONS.  33 

Ex.  6.  Differentiate  w  ==  3  sin''  x.  du  =  12  sin^  cc  cos  x  dx. 

Ex.  7.   Differentiate  u  =  cos  mx.  du  =  —  m  Binmxdx. 

Ex.  8.  Differentiate  u  =  sin  3^  cos  2^. 

du  =  (3  cos  3^7  cos  2x  —  2  sin  ^x  sin  2x)dx. 

Ex.  9.  Differentiate  u  =  sec^  ^x.  du  =  10  sec^  5a:  tan  5a7  c?a:. 

Ex.  10.   Differentiate  u  =  tan"  nx.       du  =  7i^  tan"~^  no;  sec^  no:  d'^. 

Ex.  11.  Differentiate  u  =  log  sin  x. 

Solution. — ^We  have  here  a  logarithm  to  differentiate,  {.  e.  the  logarithm  of  sin  a;. 
Hence  the  differential  is  the  differential  of  sin  x,  divided  by  sin  x,  in  the  Napier- 
ian system,  or  m  times  this,  in  the  common  system.     Therefore  du  =  — -^ •  = 

sin  X 

m  cos  X  dx 


smx 


m  cot  X  dx,  or  in  the  Napierian  system,  cot  x  dx. 


Ex.  12.  Differentiate  u  =  log  cos  x. 

du  =  —  m  tan  x  dx,  or  —  tan  x  dx. 

Ex.  13.  Differentiate  u  =  log  tan  x. 

-         sec2  X  _  dx  2dx 

du  =  ■ dx  = 


tan  X  sin  x  cos  x        sin  2x' 

Ex.  14.  Differentiate  u  =  log  cot  j;. 
Ex.  15.  Differentiate  u  =  log  sec  x. 

Ex.  16.  Differentiate  u  =  log  cosec  x.  du  =  —  cot  x  dx. 

Ex.  17.  Differentiate  u  =  e^'cos  x,  e  being  the  Napierian  base. 

Sug's.      du  =   Cdicosx)   +   cos  x  die")  =   —  e'sinxdx  +   e^cosxdx  = 
e*(cos  X  —  sin  x)dx. 

Ex.  18.  Differentiate  u  =  cce*'*"'^ 

Sug's.      du    ■=    e''°'"dx    +    xe''°^'d{cosx)    =    e^^'^dx    —    xe'^^'"' Bin  x  dx   = 
grosx^l  _  a;  sin  (c)c?a;. 

Ex.  19.  Diffsj-entiate  w  =  — ^ TXl 

du  =  e*^  sin  x  dx. 
Ex.  20.  Differentiate  u  =  log  v  sin  ^  +  log  v  cos  a;. 

Sug's.    w  =  5  log  sin  a;  +  ^  log  cos  as.     .•.(!«  =  J(cot  x  —  tan  x)dx  =  - — —. 

tan  aX 


34 


THE  DIFFEBENTIAL  CALCULUS. 


ef^ 


ILLUSTRATIVE   EXAMPLES. 

[Note. — The  object  of  these  examples  is  to  still  farther  illustrate  the  meaning  of  the  process  of 
differentiation.] 

Ex.  1.  Whicli  changes  the  faster  an  arc  or  its  sine  ?  "What  is  the 
relative  rate  of  change  ?  When  is  the  disparity  greatest  and  when 
least?  What  is  the  relative  rate  of  change  when  the  arc  is  60°? 
When  20°  ?     When  80°  ? 

SoiiTTTiON. — From  y  =  sin  a;,  we  have  by  differen-  RrT=SlP 
tiating,  dy  =  cosxdx.  The  meaning  of  this  is,  that  if 
the  arc  {x,  AP)  takes  an  infinitesimal  increment 
{dx,  Pp)  the  sine  {y,  P/)  takes  an  infinitesimal  incre- 
ment {dy,  pE)  which  is  cosjc  times  the  increment  of 
the  arc.  Now  cos  x  is,  in  general,  less  than  unity,  so 
that  the  increment  of  the  sine  is,  in  general,  less  than 
the  contemporaneous  increment  of  the  arc.  But  as  x 
grows  less  cos  x  becomes  greater  and  approaches  unity 
as  X  approaches  0.  So,  also,  cos  a?  approaches  0  as  x 
approaches  90°.  Hence  the  disparity  between  the  contemporaneous  increments 
of  an  arc  and  its  sine  is  less  as  the  arc  is  less,  disappears  when  the  arc  is  0, 
and  becomes  infinite  when  the  arc  is  90°.  For  x  =  0  we  may,  therefore ,  con- 
sider the  arc  and  its  sine  to  be  increasing  at  equal  rates.  For  x  =  90°,  the  arc  is 
increasing  infinitely  faster  than  its  sine.  When  x  =  60°  cos. r  =  ?.  Hence  at  60° 
the  sine  is  increasing  just  i  as  fast  as  the  arc.  In  the  figure,  letting  P'p'  represent 
dx,p'E.'  represents  dy  andp'E'  =  iP'p'.  "When  .a;  =  20°  cos  a;  =  .94  nearly. 
Hence  at  20°  the  sine  is  increasing  .  94  as  fast  as  the  arc.  At  80°,  the  sine  is  in- 
creasing only  about  .17 as  fast  as  the  arc.     These  facts  are  illustrated  in  the  figure. 

Ex.  2.  Assuming  that  the  relative  rate  of  increase  remains  con- 
stantly the  same  as  at  40°,  how  much  does  the  sine  increase  when  the 
arc  increases  from  40°  to  40°  10'  ?  What  when  the  arc  increases  to 
41°? 

8  14159 

StJG. — Since  the  arc  of  10'  =  — ^ =  .0029088  ;  we  find  the  increase  of  the 

180  X  6 

sine,  on  the  above  hypothesis,  to  be  .  002228 ,  which  is  slightly  in  excess  of  the  real 
increase,  as  will  be  found  by  examining  a  table  of  natural  sines  in  which  the  de- 
cimals are  extended  to  7  places.      The  table  gives  ,0022156. 

At  the  same  rate  of  increase  the  sine  of  41°  should  be  .01^9  above  the  sine  of 
40°  ;  whereas  from  a  table  the  increase  is  found  to  be  .0132714. 

[The  student  will  observe  that  the  cause  of  this  disagreement  is  that  the  rela- 
tive rate  of  increase  of  the  sine  as  compared  with  its  arc,  is  greater  at  40°  than  at 
any  point  between  40°  and  40°  10',  or  at  any  point  after  40°,] 

Ex.  3.  The  natural  tangent  of  27°  20'  is  .5168755.  '  Assuming  that 
the  relative  rate  of  increase  of  the  tangent  as  compared  with  its  arc 


TRIGONOMETRICAL  FUNCTIONS.  Zo 

remains  the  same  as  at  this  point,  for  the  next  25"  increase  of  the  arc, 
what  is  the  natural  tangent  of  27°  20'  25"  ?  Ans.,  .517029. 

Ex.  4.  Which  increases  faster,  the  arc  or  its  tangent  ?  "When  is  this 
difference  greatest?  When  least?  What  is  the  value  of  the  arc 
when  the  tangent  is  increasing  just  twice  as  fast  as  the  arc? 

Ansiver  to  the  last,  45°. 

Ex.  5.  The  natural  cosine  of  5°  31'  is  .995368.  Assuming  that  the 
relative  rate  of  change  of  the  cosine  as  compared  with  the  arc  re- 
mains the  same  as  at  5°  31',  while  the  arc  increases  to  5°  32'_,  what  is 
the  cosine  of  5°  32'?  Ans.,  .995340. 

Ex.  6.  At  36°  what  is  the  relative  rate  of  increase  of  the  arc  and 
the  logarithm  of  its  tangent? 

SuG. — From  ii,  =  log  tan  x,  we  have  du=m  {teinx-\-cotx)dx.  When  x  =  36°  this 
becomes  dit  =  .43429  X  2.102925da; ;  or  the  logarithm  of  the  tangent  increases 
about  .91  times  as  fast  as  the  arc. 

Ex.  7.  The  logarithmic  cosine  of  67°  30'  is  9.582840.  Assuming 
that  the  relative  rate  of  change  of  the  logarithmic  cosine  and  the  arc 
remains  the  same  as  at  this  point  while  the  arc  passes  to  67°  31', 
what  is  the  logarithmic  cosine  of  the  latter  arc  ?       Ans. ,  9. 582535. 

Ex.  8.  The  log  cot  58°21'  =  9.789868.     On  the  same  assumption   . 
as  above,  what  is  the  decrease  of  this  logarithm  for  1  second  increase 
in  the  arc?  Ans.,  .OOOOO'ill. 

Ex.  9.  The  log  cos  42°  14'  =  9.869474.  What  is  the  corresponding 
tabular  difference  ? 

Ex.  10.  At  what  rate  relative  to  its  velocity,  is  a  point  in  the  cir- 
cumference of  a  wheel  revolving  in  a  vertical  plane,  ascending,  when 
it  is  60°  above  the  horizontal  plane  through  the  centre  of  motion? 

Ans.,  One  half  as  fast. 


CIRCULAR   FUNCTIONS. 

'^S,  I^TOp* — The  differential  of  an  arc  in  terms  of  its  sine  is  the 
differential  of  the  sine  difjided  by  the  square  root  of  1  minus  the  square 
of  the  sine  ;  or  the  differential  of  the  sine  divided  by  the  cosine. 

Dem. — Let  y  =  sin— ^a;*,  whence  x  =  siny.     Differentiating  and  finding  the 


dx 


value  of  dy,  we  have  dy  = .     But  cos y  =  \/l  —  sin^y  =  s/l  —  x^.    .'.  dy  i= 

dx 

,  Q.  E.  D. 

\/l  —  a;2 

*  This  notation  is  explained  in  the  Trigonometry  of  this  series.    It  means  simply  "y  =  the  aro 
wh.066  Bine  is  «,  and  hence  y  =  6in""'x  is  equivalent  to  x  =  sin  y." 


36  THE  DIFFERENTIAL  CALCULUS. 

70'  ScH. — The  student  should  not  fail  to  observe  the  essential  identity 
of  this  proposition  with  [06).  Thus,  when  we  differentiate  u  =  sin  a;,  we 
get  du  =  cos  xdx,  which  expresses  the  differential  of  the  sine  (u)  in  terms 

of  its  arc  (x).     From  this  we  have  dx  = =  — ,  which  expresses 

cosa;         ^/l—u^ 

the  differential  of  the  arc  (x)  in  terms  of  the  sine  [u).     The  one  conception 
is  the  converse  of  the  other. 


77.  JPvop* — The  differential  of  an  arv  in  terms  of  its  cosine  has  the 
opposite  sign  from  the  function,  and  is  nuvfierically  equal  to  the  differential 
of  the  cosine  divided  by  the  square  root  of  1  minus  the  square  of  the  co- 
sine ;  or  the  differential  of  the  cosine  divided  by  the  sine. 

Dem. — Let  y=cos—^  x,  whence  x  =  cos  y.  Differentiating,  and  finding  the  value 

f]l*  cl'X 

of  dy,  we  have  dy  = r^ —  = .     q.  e.  d. 

^  ^  sm2/  v^l  — X-' 

7S,  ScH — Compare  this  and  the  following  propositions  with  their  equiv- 
alents in  Trigonometrical  functions,  as  was  done  in  the  case  of  the  preced- 
ing proposition. 


79.  JPvop* —  The  differential  of  an  arc  in  term.s  of  its  tangent  is  the 
differential  of  the  tangent  divided  by  1  plus  the  square  of  the  tangent. 

Dem. — Let  y  =  tan— i  cc,  whence  x  =  tan?/.      Differentiating   and  finding  the 

dx  dx 

value  of  dy,  we  have  dy  =  — ~  =  q — ■. — -,  since  sec^w  =1-1-  tan^?/  =  1  -f  x^ 
^  sec-?/       1  -[-  x^ 

Q.  E.  D. 


^0.  JPvop, — The  differential  of  an  arc  in  terms  of  its  cotangent  has 
the  02:)posite  sign  from  tlie  function,  and  is  numerically  equal  to  the  differ- 
ential of  the  cotangent  divided  by  1  idIus  the  square  of  the  cotangent. 

Dem. — Let  y  =  cot-i  cc,  whence  x  =  cot  y.    Differentiating,  and  finding  the  value 

(It  S.PO 

of  dy,  we  have  dy  =^ ^r—  =  — :; — ; — ;;.     Q-  e.  d, 

^'  ^  cosec2  y  1  j^  X' 


SI,  Pvop, — The  differential  of  an  arc  in  terms  of  its  secant  is  the 
differential  of  the  secant  divided  by  the  secant  into  the  square  root  of  the 
square  of  the  secant  minus  1. 

Dem.  — Let  y  =  sec-i  x,  whence  x  =  sec  y.     Differentiating  and  finding  the  value 


dx  dx 


'>f  dy  we  have  dy  = "- — —  = ■  since  tan  y  =  s/sec^j/  —  1  =  \/x--^  —  1. 

secy  tan  2/       ^cVx^  —  l 


Q.  £.  D. 


CIRCULAR   FUNCTIONS.  37 

S2,  Prop. — The  differential  of  an  arc  in  terms  of  its  cosecant  has 
the  opposite  sign  from  the  function,  and  is  numerically  equal  to  the  differ- 
ential of  the  cosecant  divided  by  the  cosecant  into  the  square  root  of  the 
square  of  the  cosecant,  minus  1. 

Dem. — Let  y  =  cosec-i  a;,  whence  x  =  cosec?/.     Differentiating,  and  finding  tlie 


dx  dx 


valtie  of  dy,  we  have  dy = = .  since  coty  =  \/cosec^y  —  1 

cosecycot?/  ^s/x- 1 


V  X^  1.       Q.  E.  D. 


S3,  I^TOp, — The  dfferential  of  an  arc  in  terms  of  its  versed-sme  is 
the  differential  of  the  versed-sine  divided  by  the  square  root  of  twice  the 
versed-sine  minus  the  square  of  the  versed-sine. 

Dem. — Let  y  =  Yers—^x,   whence  x  =  \ersy.     Differentiating  and  finding  tho 

(Xx 

value  of  dy  we  have  dy  =    .  '    .     But  sin  y  =  \/l  —  coss  y  =  \/l  —  (1  —  vers  y)^  = 

dx 


\/l  —  (1  — xy^  =  \/-Ax  —  a;2.     Therefore,  substituting,  dy  =  — — "- .     q.  e.  d. 

V2x  —  a;2 

S4:,  JPvop. — The  diffey^ential  of  an  arc  in  terms  of  its  coversed-sine 
has  the  opposite  sign  from  the  function,  and  is  numerically  equal  to  the 
d,ifferential  of  the  coversed-sine  divided  by  the  square  root  of  twice  the 
coversed-sine  minus  the  square  of  the  coversed-sine. 

Dem. — Let  y  =  covers—'  a;,  whence  x  =  covers  y.    Differentiating,  and  finding  the 

,        .  ,          ,         ,               cZa;  dx  dx 

value  of  dy,  we  have  ay=  —  = 


cos  y  ^i  _  gij^sy  yi  _  ^1  _  covers  s^)2 

dx  dx 

=zr.      Q.  E.  D. 


v/1  —  (1  —  xy  s/'tc  —  x^ 


EXERCISES. 

Ex.  1.  Differentiate  y  ==  sin""'^-  ;   y  =  cos~^-  ;   y  =  tan~^-  ;   y  = 

cot~^-  ;  y  =  sec~^—  ;  y  =  cosec""^-  ;  y  =  vers~^-  ;  y  =  covers"^-. 

StTG. — We  have  dy  =  — .   -■ '  .  by  (75).    Now  since  dl  -)  =  — ,  we  have 

dx 
r  dx  ^    ,.,  -  ,/  a; .  dx 


(X 
cos— ^  -  )  = 


38 


THE   DIFFERENTIAL   CALCULUS. 
rdx 

rdx 


..  =  .(t.n-.^-)  =  ^-^^;  .,  =  .(eot-.5)  =  --^,;  .,  =  .(sec- ?)  = 

rdx  /  x\  rdx  /  x\ 

—  - ;  dy  =  d{  cosec-i  -  )  = :=:z ;  dy  =  d[  vers-i  -  )  =  _- 

zVx^—r^  ^  ^^  xVx'^  —  r^  ^  rJ      v/9 

-.  ,          ,/                £c\                    dx 
and  dy  =  a(  covers—'  -  I  = 

\  ^/  ^2ra;  —  x^ 


Geometeical  IiiiiUSTBATioN. — Let  O A  =  1,  and  OA' 
=  r.  Let  y  =  arc  C  A  (to  radius  1),  and  x=  C  B'  the 
sine  of  the  same  number  of  degrees  as  y,  but  to  a  radius 


r.     Now  CB  = 


C  B 


^,  and  we  have  y  (or  CA)  = 


C  B'  V 

sin—'  C  B   =   sin—' =  sin— '  -,  the   arc    {y)   being 

taken  to  radius  1  while  the  sine  x  is  taken  to  the  radius  r. 


dx 
\/2rx  —  jc2 


B  A      B'    AT 

Fig.  15. 


Ex.  2.  Differentiate  ^  =  sin-^-  ;    ^  =  cos"^-  ;    ^  =  tan"^-  ;   ^  = 

f  IT         IT  y         V  TV 

L-1^   y         _i^   y  1^    V  1^       -.  V  ^^ 

cot  ^-  ;  -  =  sec  ^-- ;  -  =  cosec    -  ;  -  =  vers"  -  :  and  -  =  covers    -. 

f       f  T       T  TV  TV  V 


Besults  ill  order :  dy 
r^dx 


rdx 


dy 


r-  -\-  x^ 


;  dy 


\/r2  —  x^ 

r^dx 


^;  dy  = 


rdx 


\/ri 


fi x2 


-;  dy. 


r'^dx 


rdx 


s/^rx  —  x'-^ 


;  and  dy  = 


XV  X-  —  r^ 
o^dx 


;  dy  = 


r^-dx 


XV  X- 


j-2 


r2  +  072 ' 
;  dy  = 


v2rx  —  x^ 


Geometrical    L^lustbation, — Li    Fig.    15  let    OA'  =  r,    C'A'   =  y,   and 

V  X 

C'B'  =  X.     Now  if   OA  =  1,  we  have  CA  =  -,  and  CB  =  -.     Hence 

r  r 


dy  =  d(sin— '  a;)to  radius  r  = 


rdx 


v/r2  —  a;2 


r,  etc. 


85,  ScH. — The  results  in  the  last  example  will  be  seen  to  correspond  with 
those  in  Ex.  1,  by  noticing  that  in  Ux.  1  y  represents  CA,  whereas  in 
Ex.  2  it  represents  C'A'.  Now  an  increment  of  C'B'  (which  is  x  in  both 
cases)  which  makes  an  increment  in  C  A,  will  make  r  times  as  great  an  incre- 
ment in  C'A'.  Hence  we  have  but  to  multiply  the  increments  of  CA  (the 
c?y's)  as  found  in  Ex.  1,  by  r  to  get  the  corresponding  increments  in  C'A', 
which  are  the  c?y's  in  Ex.  2. 


X 


Ex.  3.  Differentiate  u  =  tan~^-. 

y 


du  = 


Ex.  4.  Differentiate  u  =  BLvr^{2x\/l  —  x^).  du  = 


ydx  —  xdy 
'Idx 


v/l  — ar» 


CIECULAR  FUNCTIONS.  39 

Ex.  5.  Differentiate  u  =  cos~\a7\/l  —  x^). 

Sug's. — By  the  rule  the  differential  of  the  arc  u  is  negative,  and  numerically 
equal  to  the  differential  of  its  cosine,  jc\/l — x^'  divided  by  the  square  root  of  1  minus 

the  square  of  its  cosine.     The  differential  of  ck  \/l  —  x~  is  dx\/l  —  x^  — 


and  1  —  {x^l  —  a;2)2  =  1  —  x2  ^  a;4  ...  ciy, 


\/l  —  x\ 
(1  —  2x2)dx 


V{1  —  x2  -f  x4)(l  —  x^) 


X  dx 

Ex.  6.  Differentiate  u  =  sin~' —  •.  du  == 


3^.37 

Ex.  7.   Differentiate  u  =  sin""^(3a:  —  4^3).  du  =  -- 

V   1  ^2 

Ex.  8.  Differentiate  u  =  vers"^?/  —  \/2r2/  —  y^,  understanding  that 
vers~^?/  is  taken  to  radius  r. 

SuG.     dw  =         ^^1         _   rdy—ydy   ^        ydy 

s/'lry  —  y-        \/2ry  —  y~        \/2ry  —  y^ 


Ex.  9.  Differentiate  u  =  tan~^(\/l  +  x'^  —  x). 

du  = 

Ex.  10.  Differentiate  u  =  log  a^/y— ^- 1-  i- tan~^a;. 


dx 


2(1  +  ^=)' 


dr 


SuG.     w  =  4  log  (1  +  as)  —  4  log  (1  —  ic)  +  ^  tan-»  x.    du  =  — £- 


fl;4 


mdx 
Ex.  11.  Differentiate  v  =  sin~'ma;^  —  -  cZv  =     /         — = 

^        vl  —  m2^-5 

Ex.  12.  Differentiate  y  =  e»'*'^~\  tZy  =  e*"~'"    ^"^ 


1  +  ^■^' 


Ex.  13.  Differentiate  y  =  tan""^^ ^ — .  dy 


1  —  x^'  1  +  x^' 

Ex.  14.  Differentiate  ?/  =  x^^''~\ 

1 

dy  ==  x'^''    ^  :j — ^i— ^ ^ —  Y  dx. 


x{l  —  x^) 


GENERAL    SCHOLIUM, 

86,  The  preceding  sections  comprise  the  fundamental  rules  of  the  differ- 
ential calculus  ;  and  it  only  remains  to  extend  and  apply  them,  in  order  to 
complete  this  portion  of  our  subject. 


40 


THE    DIFFERENTIAL    CALCULUS. 


SUCTION    IV, 

Successive  Differentiation  and  Differential  Coefficients. 


SUCCESSIVE    DIFFEKENTIATIOK 

87 »  Bef. — Successive  Differentials  are  differentials  of  dif- 
ferentials ;  or  a  successive  differential  is  the  difference  between  two 
consecutive  states  of  a  differential. 

III. — Let  M  N  Fig.  16,  be  a  straight  line  whose 
equation  is  y  ^=^  ax  -\-  b  ;  whence  dy  =  adx.  Now 
suppose  X  to  be  considered  equicrescent,  and  let 
DD',  D'D",  D"D"',and  D"'D''^  represent  the 
successive  equal  increments.  P'E,  P"E',  P"'E", 
and  P'vE'"  represent  the  contemporaneous  incre- 
ments of  y,  i.  e.  the  dy's.  But  in  this  case  the  dy's 
are  all  equal.  Hence  there  being  no  difference  be- 
tween two  successive  states  of  dy,  as  between  P'E 
and  P"E',  there  is  no  successive  differential,  or  the 

differential  of  dy  is  0,  since  dy  is  constant.  This  fact  appears  also  from  the  rela- 
tion dy  =  adx,  in  which,  if  we  conceive  dx  to  be  constant  (i.  e.,  x  equicrescent), 
adx  is  constant ;  whence  dy,  which  equals  adx,  is  constant. 

But  consider  in  a  similar  manner  the  parabola  in  Fig.  17, 


Fig.  16. 


whose  equation  is  y'^ 


2px  ;  whence  dy  =^— . 


Still  con- 


FiG.  17. 


sidering  dx  as  constant,  i.  e.  DD'  =  D'D"  =  D"D"' 
=  D"'D'^,  etc.,  it  is  evident  that  the  dy's,  which  are 
represented  by  P'E,  P"E',  P"'E",  etc.,  are  not  constant. 
Now  the  difference  between  any  two  successive  values  of 
dy,  as  between  P'E  and  P"E',  is  a  successive  differ- 
ential, i.  e.  a  differential  of  a  differential.  The  fact  that 
dy  is  a  variable  in  this  case  when  dx  is  constant  is  also 

Tfdx 
readily  seen  from  its  value  dy  =- — .     In  this  pdx  is  constant,  but  y  is  variable. 

Hence  dy  varies  inversely  as  y. 

88.  Def. — A  Second  Differential  is  a  differential  of  a  first 
differential,  is  represented  by  d^y,  and  read  "  Second  differential  y." 
A.  Third  Diff'erential  is  a  differential  of  a  second  differential, 
is  represented  by  d?y,  and  read  "  Third  differential  y."  In  like  man- 
ner we  have  fourth,  fifth,  etc.,  differentials. 

ScH. — The  student  should  be  careful  not  to  confound  d^y  with  dy'^.  The 
latter  is  the  square  of  dy.  Nor  should  the  superior  2  in  d'^y  be  mistaken 
for  an  exponent :  it  has  no  analogy  to  an  exponent.     Observe  the  significa- 


SUCCESSIVE  DIFFERENTIATION.  41 

tion  of  the  several  expressions  d'^y,  dy^  and  d{y'^).     The  latter  is  equivalent 
to  lydy. 

89,  JProp, — Second  differentials  are  formed  by  differentiating  first 
differentials,  third  differentials  by  differentiating  second  differentials,  etc., 
according  to  the  rules  already  given. 

This  proposition  is  self-evident,  since  the  differentials  are  expressed  as  algebraic, 
trigonometric,  logarithmic,  or  exponential  functions,  the  rules  for  differentiating 
which  are  those  heretofore  given. 

Ex.  1.  Produce  the  several  successive  differentials  of  y  ==  ax*. 

Solution. — Differentiating  y  =  ax\  we  have  dy  =  4:ax^dx.  Differentiating  this 
differential  remembering  that  d{dy),  i.  e.  the  differential  of  dy  is  written  d~y,  and 
that  dx  is  constant,  we  have  d-y  =  12ax'^dx  dx,  or  12ax^  dx-.  In  like  manner  dif- 
ferentiating d'^y  =  VlaxHx"^,  we  have  d'^y  =  24«a;  dx? .  And  again  d'*y  =  2^adx*. 
Here  the  operation  terminates,  since  d'^y  being  equal  to  24«dx-*  is  constant. 

Ex.  2.  Produce  the  several  successive  differentials  oi  y  =  Sx*  — 
dx^  —  5x. 

'  dy  =   (32^3  —  9^i2  —  h^dx. 

Results,  - 


dHj  =  (96^2  —  18^)^^% 
dHj  =   (19207  —  l%)dx\ 
d^y  =  ld2dx^ 


Ex.  3.  Produce  the  first  six  successive  differentials  oi  y  =  sin  x. 

r  dy  =  cos  x  dx,  d^y  =  • —  sin  x  dx^, 
Results,   \  d^y  =  —  cos  x  dx^,  d^y  ==  sin  x  dx*, 
[  d^y  =  cos  X  dx-',  d^y  =  —  sin  x  dx^. 

QuEBY. — Does  the  above  process  ever  terminate  ? 

Ex.  4.  What  is  the  3rd  differential  oi  y  =  af  ? 

d^y  =  n{n  —  l)(n — ■1)x''~^dx^. 


Ex.  5.  Produce  the  4:th  differential  oi  y  =  ax^. 


15a  dx* 


Ex.  6.  Produce  the  first  six  successive  differentials  of  y  =  cos^. 

Ex.  7.  Produce  the  first  four  successive  differentials  oi  y  =  logx, 

in  the  common  system. 

^      ,       ,         mdx    ^  m  dx^    ,  2m  xdx^      2m  dx^    _ 

Results,  dy  = ,  d^y  =  — ,  d^y  = = ^,  d*y  = 

'     ^  X         ^  x^         ^  X*  x^  ^ 

6??i  dx* 

X* 


42  THE  DIFFERENTIAL  CALCULUS. 

Ex.  ^  Produce   the  first    four    successive    differentials   oi  y  =: 

log  (1  +  ^)>  in  the  connnon  systera. 

^      ,       ,  mdx    ^  m  dx^     ,         ImilA-x)  dx"^       2m  dx^ 

Results,  dy  = ,  d^y  =  — — ,  d^y  =  — y-—^ — '- = , 

^        l^x      ^  {1+xy     ^  {1  +  xy  {l-{-xy 

,  Qmdx* 

Ex.  9.  Produce  the  fourth  differential  oi  y  =  ef. 

d^y  =  e'dx*. 

Ex.  10.  Produce  the  fourth  differential  oi  y  =  a^,  in  the  common 

-          log^a     , 
system.  dni  = a'^dx*. 


DIFFERENTIAL    COEFFICIENTS. 

90,  Defs. — A  First  JDifferenticd  Coejficient  is  the  ratio  of 

the  differential  of  a  function  to  the  differential  of   its  variable,  and  is 

dy 
represented  thus,  -7^,  y  being  a  function  of  the  variable  x. 

A   Second  Differential  Coefficient  is  the  ratio  of  the 
second  differential  of  a  function  to  the  square  of  the  differential  of 

its  variable,  and  is  expressed  thus,  -r-^. 

A  Third  Differential  Coefficient  is  the  ratio  of  the  third 

differential  of  the  function  to  the  cube  of  the  differential  of  its  vari- 

d^y 
able,  and  is  represented  thus,  -7^.     In  Hke  manner  the  nth  differen- 

(XX 

d^y 

tial  coefficient  is  -7—. 

dx" 

S/u  civ 

111. — Having  y  =  ax\  we  obtain  --  =  5ax^.    In  strict  propriety  —  is  a  symbol 

representing  the  general  conception  of  the  ratio  of  an  infinitesimal  increment  of 
the  function  to  the  contemporaneous  infinitesimal  increment  of  its  variable  ;  and 
5ax*  is,  in  this  case,  its  value.  But  it  is  customary  to  speak  of  either  as  the  dif- 
ferential coefScient.  The  appropriateness  of  the  term  differential  coefficient  arises 
from  the  fact  that  the  Sax'  is  the  coefficient  by  which  the  differential  of  the  vari- 
able has  to  be  multiplied  in  order  to  produce  the  difi'erential  of  the  function. 
Strictly,  therefore,  the  differential  coefficient  is  the  coefficient  of  the  differential 
of  the  variable  ;  but  it  is  customary  to  speak  of  it  as  the  differential  coefficient 
of*  the  function. 


*  The  "of"  meaning,  perliaps,  "derived  from,"  or  "appertaining  to." 


DIFFERElilTIAL   COEFFICIENTS.  43 

Ex.  1.  Given  y  ■=  ax^  —  x^,  to  find  the  1st,  2nd,  and  3rd  differential 
coefficients. 

Results,  —  =  3ax^  —  2x,  — ^  =  6ax  —  2,  -— ^  =  6a. 
ax  ax'^  dx^ 

1  -\-  X 

Ex.  2.  Given  v  =  z ,  to  find  the  5th  differential  coefficient. 

.         1  —  X 

dni  240 


dx'^        (1  —  a;)« 
Ex.  3.  Given  y  =  <f ",  to  find  the  4th  differential  coefficient. 

dx-^ 
Ex.  4.  Given  y  ==  a%  to  find  the  5th  differential  coefficient. 

clHl  . 

—^  =^  log^aa, 

Ex.  5.  Produce  the  first  three  successive   differential  coefficients  of 
y  =  tan  x. 

-r.    7     dy  d-y      r.         J         d^y       i         x 

Results,  -^  =  sec2j-,    -^  =  2  sec^  j;  tan /c,    -7—   =  4  sec^  a;  tan^  j;  + 
'  dx  dx'  dx^ 

2  sec^  X. 

[Note. — The  10  examples  in  the  former  part  of  this  section  may  be  vised  for  further  illustration 
of  this  subject,  if  desired.] 

Ex.  6.  Given  y^  =  2px,  to  form  the  thu'd  differential  coefficient. 

dv       T) 
SoiiiJTioN. — Diflferentiatinar  and  findinsr  the  coefficient,  we  have  --  =  -.     To  dif- 

^  ^  dx       y 

dv 
ferentiate  ---  we  have  but  to  remember  that  dy  is  a  variable,  that  its  differential  is 
do; 

dhi,  and  that  dx  is  constant  and  hence  remains  the  same  ;   whence  d(  ---)  =  — . 
^  \dx/         dx 

Differentiating  -,  we  have  d(~]  =  —  — -.    Hence  — ^  =  —  --^.     But  the  second 

y  \y/  y-  dx  2/' 

differential  coefficient  is  the  ratio  of  the  second  differential  of  the  function  to  the 

d  v 
square  of  the  differential  of  its  variable.     Hence  dividing  by  dx,  we  have  — ^  = 

dy  p 

P~^~  P 

dx  dy      p  d^y  V  v^ 

—  —— .  But  --  =  -.     Substituting  this  value,  -^^  = =  —  — .     In  a  similai 

2/2  dx      y  .  dx^  y^  y^ 

.    ,  d^j       3p3 
manner  we  find  -r—  =  — .  ^ 

dx^       y= 

d^v  d^v  R* 

Ex.  7.  From  A-y^  +  R"-x^  ==  A^R%  find  ~.  '^  — 


dx^  dx^  A'-y^ 


d^if       2c^ 
Ex.  8.  From  xy  =  c^  show  that  -—  =  — . 
^  dx^        x^ 


4:4  THE   DIFFERENTIAL    CALCULUS, 

Ex.  9.  From  y  = show  that  — -  = 


1  —  X  dx*        (1  —  xY' 

Ex.  10.  From  y^  =  sec  1x  show  that  i/  +  -^^  =  3y5. 

[Note. — The  first  diflferential  coefficient  expresses  the  relative  rate  of  increase  of  the  function 
and  its  variable,  and  its  significance  is  illustrated  in  the  examples  on  pages  34,  35,  which  it  wiU  be 
well  to  review  for  this  purpose.] 

dxi 
Ex.  11.  "What  is  the  third  differential  of  w  =  ■—,   dx  being  con- 

dX 

stant?  d^u  =  --. 

dx 

-  dv 

Ex.  12.  "What  is  the  differential  of  --,  dx  not  being  considered 

dX 

,  -  .        d^ydx  —  d'^xdy. 

constant?  ♦  Ans., 

dx^ 

ScH. — ^Differential  coefficients  are  generally  variables  and  hence  can  be 
differentiated.  In  differentiating  them  it  is  important  to  observe  whether 
dx  is  conceived  as  constant  or  variable.  To  say  that  the  first  differential 
coefficient  is  generally  variable  is  equivalent  to  saying,  in  the  case  of  a  curve, 
that  the  relative  rate  of  change  of  the  ordinate  and  abscissa  varies  for  dif- 
ferent points,  as  we  have  seen  heretofore. 


^♦» 


SECTION  V, 

Functions  of  Several  Variables,  Partial  DiiGFerentiation,  and 
Differentiation  of  Implicit  and  of  Compound  Functions. 

01,  "When  a  quantity  is  a  function  of  two  or  more  variables,  as 
u  =f{x,  y),  these  variables  may  be  independent  of  each  other,  or 
dependent  upon  each  other.  This  distinction  is  marked  by  saying 
that  14  is  a  function  of  two  or  more  independent,*  or  two  or  more 
dependent*  variables  as  the  case  may  be. 

IiiL. — Let  X  and  y  be  th.e  sides  of  a  rectangle  and  u  its  area.  Then  u  =  xy,  in 
which  w  is  a  function  of  the  two  independent  variables  a;  and  y.  That  is,  x  may 
vary  without  causing  any  variation  in  y,  and  y  may  vary  without  causing  a  varia- 
tion in  X. 

But  suppose  u  to  represent  a  rectangle  which  is  to  remain  similar  to  a  given, 
rectangle.  Now  when  x  or  y  changes,  the  other  must  change  also,  and  we  have  n 
a  function  of  two  deper^ient  variables. 

*  These  terms  are  here  used  in  a  slightly  different  sense  from  that  hitherto  assigned  them. 


PABTIAL  DIFFERENTIAIION.  45 

92,  Def. — A  I^artial  Differential  of  a  function  of  two  or 
more  variables  is  a  differential  taken  under  the  hypothesis  that  one 
or  more  of  the  variables  remains  constant ;  usually  we  consider  the 
partial  differential  under  the  hypothesis  that  but  one  of  the  variables 
changes,  the  others  remaining  constant. 

03.  Def. — A.  Total  Differential  of  a  function  of  two  or 
more  variables  is  a  differential  taken  under  the  hypothesis  that  all  the 
variables  upon  which  it  depends  vary. 

III. — If  u  =  Sax^y  —  2y^  -f~  ^^^^  —  ^'  ^^^  ^  ^^^  V  ^^"®  conceived  as  entirely 
independent  of  each  other,  *  it  is  evidently  legitimate  to  inquire  what  effect  upon  u 
will  be  produced  by  a  change  of  x  or  y  alone.  Thus,  if  we  suppose  x  to  take  an 
infinitesimal  increment  and  y  to  remain  unchanged,  the  contemporaneous  incre- 
ment of  tt  is  dw  =  {Gaxy  -j-  Qbx^)dx.  If,  on  the  other  hand,  y  takes  an  infinitesi- 
mal increment  and  x  remains  constant,  the  contemporaneous  increment  of  u  is 
du  =  (Sax-  —  4:y)dy. 

But  how  is  it  when  x  and  y  are  dependent  upon  each  other  ?  Evidently,  if  we 
wish  to  obtain  the  total  change  in  u  due  to  a  change  in  both  x  and  y,  we  may  still 
conceive  them  to  change  m  succession.  Only,  in  this  case,  we  must  remember  that 
the  change  in  y  (for  example)  is  not  independent  of  the  change  in  x.  That  is, 
that  dy  is  a  function  of  dx,  which  is  not  the  case  under  the  former  hypothesis. 

94,  Def. — A    Partial    Differential    Coefficient  is  the 

ratio  of   a  partial  differential  of   a  function  of  several  variables,  to 
the  differential  of  the  quantity  supposed  to  vary. 

95,  Def. — A  Total  Differential  Coefficient  of  a  func- 
tion of  two  or  more  variables  is  the  ratio  of  the  total  differential  of 
the  function  to  the  differential  of  some  one  of  the  variables  ;  and 
there  may  be  as  many  such  coefficients  as  there  are  variables. 

96,  ScH. — When  several  independent  variables  enter  into  a  function,  we 
might,  if  we  chose,  consider  each  of  them  equicrescent,  and  its  differential 
constant ;  but  when  the  variables  are  mutually  interdependent,  it  is  evident 
that  in  general  but  one  can  be  considered  equicrescent. 


97,  I^vop, — The  total  differential  of  a  function  of  several  variables 
is  equal  to  the  sum  of  the  partial  diffci'entials. 

Dem. — Let  u  =/(iC,  y).  Kepresent  the  partial  differential  of  u  with  respect  to 
X,  by  dxU  and  with  respect  to  y  by  dyU,  while  du  represents  the  total  differentiaL 
Now  du  =J\x  -\-  dx,  y  -\-  dy)  — f{x,  y)  ;  i.  e.  it  is  the  difference  between  two  con- 
secutive states  of  the  function  when  both  x  and  y  have  taken  increments.  Now 
subtract  and  add  f{x  -f-  dx,  y),  which  will  not  change  the  value,  and  we  have 

du  =f{x  +  dx,  y  +  dy)  —fix  -f-  dx,  y)  +/(a;  -f  dx,  y)  —f(x,  y). 
T^ViifyX  -f-  dx,  y  -f-  dy)^  — f(^x  -\-  dx,  2/)+  =  dyU,  since  it  is  the  difference  betweeu 

*  SometMng  in  the  nature  of  the  problem  always  determines  whether  x  and  y  are  dependent  or 
independent. 

t  The  student  will  notice  that,  however  x  and  y  are  involved,  the  only  difference  between 
J\x  -\-  dx,  1/  -j-  dy)  and  f{x  -{-  dx,  y)  is  what  arises  from  y  passing  to  y  -\-  dy. 


46  THE   DIFFERENTIAL   CALCULUS. 

two  consecutive  states  of  tlie  function  due  to  a  change  in  y  alone.  So,  also, 
f{x  +  dx,  y)  —  f{x,  y)  =  dxU,  for  a  like  reason.  .  • .  du  =  dxU -j-  dyU  ;  and  it  is 
evident  that  a  similar  course  of  reasoning  can  be  applied  when  w  is  a  function  of 
any  number  of  variables,     q.  e.  d. 

III.— This  proposition  may  be  perplexing  to  a  thoughtful  student,  even  after 
he  has  learned  the  demonstration.  The  following  illustration,  for  the  substance 
of  which  I  am  indebted  to  Price,  may  help  him  to  reahze  its  truth.  Consider  the 
amount  of  grain  grown  upon  a  piece  of  land.  This  is  evidently  a  function  of  the 
area,  the  soil  (quality  of),  and  the  Ullage,  to  say  nothing  more.  Let  G  represent 
the  total  amount  of  grain,  A  the  area,  S  the  soil  (quality  of),  and  T  the  tillage. 
In  mathematical  notation  we  have  G  =f{A,  8,  T).  Now  consider  A,  S,  and  Tas 
variable. 

1st.  "We  may  inquire  what  effect  upon  G  will  be  produced  by  an  increment  (for 
our  present  purpose  infinitesimal)  of  A,  while  S  and  T  remain  constant.  This 
will  give  a  partial  differential  of  6^  with  respect  to  A.  In  like  manner  we  may 
inquire  what  effect  upon  G,  an  infinitesimal  increment  of  S,  or  infinitesimal  incre- 
ments of  A  and  S,  will  produce.  And  again  what  effect  will  be  produced 
by  a  change  in  any  one,  any  two,  or  in  all  three  of  the  variables.  A,  8,  and  T. 
The  latter  would  be  the  total  differential  of  G. 

2nd.  It  is  evident  that  A  is  independent  of  8,  and  T,  while  8  and  T  are  (proba- 
bly) dependent  upon  each  other,  i.  e.  good  tillage  may  have  a  larger  proportionate 
effect  upon  the  crop  on  good  soil,  than  upon  poor. 

3rd.  If  we  consider  the  effect  upon  G,  produced  by  an  infinitesimal  increment 
of  A,  while  8  and  Tare  constant,  calling  this  d^(r  ;  and  then  (not  considering  A 
as  having  increased)  consider  the  effect  on  G  of  an  infinitesimal  increment  of  8, 
calling  it  dsG  ;  and  then  (not  considering  either  J.  or  iS  as  having  increased)  con- 
sider the  effect  upon  G  produced  by  an  infinitesimal  increment  of  T,  calling  it 
drG  ;  it  may  seem  that 

dG  =  dAG  -f  dsG  4-  drG 
will  not  represent  the  total  change  in  G  when  A,  8,  and  T  change  together.  For 
example,  we  lose  the  effect  of  the  improvement  in  soil  and  tillage  upon  the  incre- 
ment of  the  area,  and  also  the  effect  of  the  increased  effect  of  better  tillage  upon 
the  improvement  of  the  soil.  But  it  is  easy  to  see  that  these  effects  are  infinites- 
imals of  infinitesimals,  and  that  the  effects  we  seek  to  trace  are  infinitesimals  of 
the  first  order,  hence  the  former  must  be  omitted. 

Anothee  Illustration  to  the  same  purpose  is  furnished  /j /L 

by  the  paraUelopipedon.     Let  u  represent  the  volume  of       //  \ 0'/^ 

the  paraUelopipedon    indicated   by  the  dotted  lines,  of    A       r  — A~    i 

which  X,  y,  and  z  are  the  length,  breadth,  and  height,        j    /'  1    /y 

respectively.     Then  it  = /(cc,  ?/,  z)  =  a;?/^.     The  additions     ^- - -)f^ 

represented  upon  the  end,  sid%  and  top  respectively,  are  Yiq  18. 

dxU,  dyii,  and  d^u,  but  it  does  not  appear  that  the  sum  of 

these  make  up  the  total  increment  of  u  due  to  an  increase  of  all  three  of  the  vari- 
ables. But  it  does  appear  that  the  wanting  parts  will  be  infinitesimals  of  the 
second  and  third  orders,  when  the  increments  of  x,  y,  and  z  are  infinitesimal,  and 
as  the  increments  represented  in  the  figure  are  infinitesimals  of  the  first  order, 
the  others  must  be  omitted,  in  relation  to  the  latter. 

Ex.  1.  On  the  principle  of  the  above  proposition  dijfferentiate  u  = 


PARTIAL   DIFFERENTIATION.  47 

3a;2i/2 —  5^3 —  2?/  — 10.     Observe  also  that  the  result  agrees  with  that 
obtained  by  the  method  before  learned. 

Solution. — Differentiating  with  respect  to  x,  we  have  dxU  =  Qxy^dos  —  15x'^dx. 
Again,  differentiating  with  respect  to  y,  M'e  have  d,/U  =  6x"ydy  —  2dy.  Adding, 
du  ==  d^u  4-  dyU  =  Gxy-dx  —  15x-dx  -\-  Gx-ydy  —  2dy.  Finally,  differentiating  by 
the  elementary  methods,  du  =  6xy'dx  -I-  6x*ydy  —  15x'^dx  —  2dy,  a  result  identi- 
cal with  the  preceding. 

Ex.  2.  Differentiate  u  =  x'-'  both  by  the  above  principle  and  by 
passing  to  logarithms  and  using  the  elementary  methods,  and  com- 
pare the  results. 


Sug's.     dxU  =  yx'J—^dx.      d,ju  =  x^ log x dy.      .-.    du  =  yx^—'^dx  -j-  x^logx dy. 
gain 

x'-iydx 


fill           cloc                                     li/ncLo^ 
Again  log  u  =  y\ogx.     Hence  —  =  y \-\ogxdy,  oxdu  = \-u\o%xdy  = 

XL  QC/  X 


-f-  ^^  log  xdy  =  x'J-'^y  dx  -{-  x^  log  x  dy. 


y 

Ex.  3.  Differentiate  14=  tan~^-,  both  by  the  method  of  partial  differ- 
entials and  by  the  elementary  method. 

ydx  dy 

SuG  s. — By  partial  differentiation,  dxU  =  = —, — •.     dyU  =  — —  = 

/y\      xdy— ydx 


<f) 


xdy              ,        xdy — ydx     .^    ,,       ,          ,             ,•,-,-,            v-^ 
— ; — .    .  • .  cm  =  — =^— .    By  the  elementary  method,  aw  = 

a;2  +  2/2  a;2^_2/2  ^  ^  ^  y-2  ^  y^ 

x^  '"" 

xdy  —  ydx 

x'^  4"  2/' 

[Note. — The  pupil  will  doubtless  be  led  to  inquire,  "Why  use  the  method  of  partial  differentiation 
when  tLe  elemeutai'y  methods  seem  to  be  more  expeditious?  It  is  not  for  its  use  in  such  ele- 
mentary processes  that  it  is  valuable.  These  examples  are  given  only  to  illustrate  the  propo- 
sition ;  the  utility  of  it  will  appear  hereafter.] 

X  -4-  y 
Ex.  4.  Differentiate  as  above  u  = — . 

jc  —  ij 

Sug's      d.u  =  (^~2/)c^^—  (x-^y)dx  ^  —2ydx      ^  ^^  ^  {x  —  y)dy  ■\- {x -{- y)dy 

{x  —  yy^  {x—yy'       "  («;  —  2/)^ 

2xdy  ,        2{xdy  —  ydx) 

—  •    du- — 


{x—yY  K^  —  yy 

Ex.  5.  Differentiate  as  above  u  =  sin  (xy). 

du=  cos  (xy)[ydx  -f  xdy], 

Ex.  6.  Differentiate  as  above  u  =  logx'\ 

ydx 
du  ==  d^u  +  dyii  =  ■ \-  log  X  dy. 


48  THE  DIFFERENTIAL  CALCULUS. 

Ex.  7.  Differentiate  as  above  u  =  y'*"''. 

du    =    d^u    +    d,u    =    y'^'^'logycosx  dx   +    sin  xy'^'^'-^dy    = 

dy 
y''^  '  log  y  cos  xdx-]r&ui  x-—^^. 

y 

X 

Ex.  8.  Differentiate  as  above  u  =  vers"^-. 

y 

X  dy 
dx  y ydx  —  xdy 

V'lxy  —  ^2        \/2xy  —  x^        yv2xy  —  x'^ 

Ex.  9.  Differentiate  as  above  u  =  sin  {x  -\-  y). 

du  =  cos  {x  -\-  y)  [dx  +  dy], 

Ex.  10.  Differentiate  as  above  u  =  ^2^222. 


98,  dotation, — Since  when  u=f{x),  du=  the  first  differential 
coefficient  of  u  with  respect  to  x,  multiplied  by  dx,  we  may  symbolize 

dM,  by  -j-dx.     So  also  dyU  =  -j-dy. 
'  '     ^  dx  dy  -" 

09,  Prop, — The  formula  for  the  total  differential  coefficient  of  U 
with  respect  to  x,  when  u  ==  f (x,  y)  is 

tdu-i,  du  du  dy 

dxA  dx  dy  dx* 

in  which  —   and  t—  are  partial  differential  coefficients,  and  the  [  1  indi- 
dx  dy 

cate  the  total  coefficient. 

3m  du 

Dem.— We  have  du  =  d^u  -\-  dyU  =i  —  dx  -{-  -j^V-     Dividing  by  dx,  and  dis- 

/v.  .     ■  ■.      ■!■--■         1         VdvTK       du  ,  du  dij 
tinguishing  the  total  differential  coefficient  by  the  [  J,  we  have     --==--  -j-  -  -  •--. 

Q.   E.    D. 

ScH.  — This  f  ormTila  has  definite  meaning  only  when  y  and  x  are  mutually 

dy 
dependent,    i.  e.  when  y  =/(.r),  since  otherwise  — -  is  indeterminate.     The 

uoo 

formula,  therefore,  signifies  that  when  u  =/(.r,  y),  and  y  =  fi{x),  the  total 
differential  coefficient  of  u  with  respect  to  x  is  equal  to  the  partial  differ- 
ential coefficient  of  u  with  respect  to  x,  +  the  product  of  the  partial  differ- 
ential coefficient  of  u  with  respect  to  y,  multiplied  by  the  differential  co- 
efficient of  y  with  respect  to  x,  obtained  from  the  relation  y  =f^[x). 

The  following  language  is  often  used  to  express  the  relation  of  u  to  x  and 
y  in  such  a  case  ;  viz. ,  u  is  directly  a  function  of  x,  and  indirectly  a  function 

of  X  through  y. 

rdirx        du        du  dy  ,  . 

It  might  seem  that  the  relation     —  I  =  -; — I — ,-  -r  is  absurd,  smce  by 

LdvA        dx        dy  dx 


FUNCTIONS  OF  SEVERAL  VARIABLES.  49 

tdu-\         du  , 

—    =  2 — .     But  this  IS  to  misapprehend 

entirely  the  signiiGlcance  of  the  notation.     It  is  to  be  observed  that  the  du 

in  — ,  is  by  no  means  necessarily  the  same  as  the  du  in  --,  or  in  I  --   .     In 
d.v  J  ^  cLy  \_dx\ 

dtt 

— ,  du  is  the  increment  of  u  due  to  an  infinitesimal  increment  of  x  in  the 
dx 

function  u  =f{x,  y),  while  y  remains  constant.     In  like  manner  the  du  in 

(I'll 

— ,  is  the  increment  of  u  due  to  an  infinitesimal  increment  of  y,  while  x 
dy 

remains  constant.     These  will  by  no  means  be  generally  the  same.     Much 

less  will  either  of  these  c?w's  be  the  same  as  the  du  in    --  1,  which  is  the 

\_dxj 

change  in  u  due  to  a  change  in  both  the  variables  x  and  y.     Finally,  there 

,..,,,.„,,  ,         -,.,,■,  .       du  dy 

IS  nothing  in  the  logic  of  the  process  by  which  the  expression  —  -r-  was 

cfu  exec 

arrived  at,  that  makes  the  (iy's  in  it  equal  to  each  other.     The  dy  in  —  is 

simply  an  arbitrary  infinitesimal  increment  assigned  to  y  in  u  ==/{x,  y),  x 

dy 
remaining  constant ;  while  dy  in  --  is  the  increment  which  y  takes  on  in 

(jLOO 

the  function  y  =  f^{x),  when  x  takes  the  increment  dx. 

^^^    ^       w       nnr  «/  V        ,        r<5un        du       du  dy 

100.  Cor.  1.— From  u  =  f  (x,  y,  z)  we  have  |^— J  =  —  +  _  ^  + 

— -  -tp-,  in  which  y  and  z  are  functions  of  x. 
dz  dx 

Dem.— Since  the  total  differential  equals  the  sum  of  the  partials,  we  have 

du  ^      ,    du  ^      ,    du ,        ^.  .  ^.       ,       -,      rdul        du    ,    du  dy    ,    du  dz 

<'"  =  dx'^  +  df^  +  df'-    ^™*"^s  ty  cte,  |_--J  =  -  +  --  ^  +  -  -. 

Q.  E.  D. 

ScH. — In  the  same  manner  the  total  differential  of  a  function  of  any 
number  of  variables  dependent  or  independent,  may  be  found  ;  and,  when 
all  are  dependent  upon  some  single  variable,  the  differential  coefficient 
with  respect  to  that  one  may  be  formed. 

101,  Cor.  2. — If  u  ==  f(y,  z,  w),  and  y  =  ^(^),  z  =  (px{^)>  fl^w^Z 

[Let  the  student  give  the  proof.] 

tjc  \~dw~\ 

Ex.  1.  Given  u  =  tan~^-,  and  y^  -{-  x^  =  r",  to  find    --  . 

c  TTT    1  r<^w"l        <^w    ,    du  dy     ^  ,..-,,  du       ^  du 

Solution. — VVe  have     --     ==  - — L  _-  -Ji.     Remembenng  that  ^-  and  --  are 
LaxJ        dx        dy  dx  <=  ^  ^y 


50  THE  DIFFEEENTIAL  CALCULUS. 

dx 


partial  differential  coefacients,  we  have  from  u  =^  tan-^-,  du  =  bsbbee 


/acx  dx 


2/' 
ydx                 du     y    ^    ^.^                       ^    ^du         x      , 
=  -^  3  wnence  — =  - .  In  like  manner  we  find  — = ^.  Also  from  y^-f-a;? =r2, we 

find  ^-^  =  —  -.     Substituting  these  values,  we  obtain  f--!  =  ^  4-  ^  —  -  Y—  -\ 
dx  y  ^  \_dx_\      r2^\      rVV      y). 

_  2/    ,   jK^  _  y^  +  x-2  _  r2  _  1  1 

r^       r  ?/  ~      r2y      ~  r-y  ~  y  ^^  ^^2  _  ,^' 

Ex.  2.  Given  w  =  iQjr^{xy),  and  y  =:  e*  to  form  f— 1. 

V.dxi 

BesuU,  r^l  =  f:(l+f2. 
LdxA        1  +  ^2e'^ 

Ex.  3.  Given  u  =  z"-  -\-  y^ -{■  zy,  and  z  =  sin  ^,  y  =  ^>  to  form  f— 1- 

Ldxj 

SxTG's.-We  have  [--1  = --  ^  +  -  - 
LdxJ        fZ^/  dx        dz  dx 

=  (32/2  _j_  2)ex  _^  (2z  4-  2/)  (cos  a;) 

=  (3e2^  -[-  sin  .-rje^^  j-f-  (2  sin  a;  -f  e^)(cosa;) 

-=  Se^^^  -j-  6^(sina;  -f-  cosic)  +  2  sin  jc  cos  a; 

=  3e3==  -]-  e^(sin  ic  -f-  cos  a)  -|-  sin  2x. 

Ex.  4.  Given  w  ==  yz,  and  y  =  ^,  z  ==  ^-^  —  Ax^  +  12j;2  —  24^  +  24, 
to  find  g].  Sesnlt,  g^]  =  e-... 

Ex.  5.  Given  u  =  sin-^(^ —  q),  and^=  Sx,  q=:4:X^,  to  find  f-^l. 

c     '       xtr     T.         <^'^                    1                   du                    —  1  dp 

buGs. — We   have   —  =  —  ^      —    =    —  .     -£-   =   3 

«/>         v/1  —  {p  —  5)2       Q'?           v/1  —  (p  —  g)2      t^^ 
and  1^    =    12^2.        Whence    Pill    =    ^         _    _  ^^:^^ 

"•^  LdxJ  ^1    __    (p   _   g)2  v/l    _    (p    _    ^j2 

3  -^  12.r2  3 


v/1  —  9a;2  +  24x4  —  16x6         v^l  —  x-^ 

Ex.  6.  Given  li  =  — p- ^  +  ^  and  2/  =  log^,  to  find  that  r— 1 

=  a:3(log;r)2. 

Ex.  7.  Given  w  = --^,  where  p  =  a  sin  a:,  and  q  ==  cos  a;,  to 

find  that  I— 1  =  e^sina;. 

[Note.— In  such,  examples  as  the  above,  it  is  of  course  possible  to  substitute  in  the  ftinction  u, 
the  values  of  the  several  variables  on  wbicb  it  depends,  in  terms  of  the  single  variable  upon 


FUNCTIONS   OF  SEVERAL  VARIABLES.  61 

which  each  of  them  depends,  and  then  have  m  =  a  function  of  a  single  variable,  which  can  be 
differentiated  by  the  elementary  processes.  But  it  is  the  chief  design  of  these  examples  to  far 
miharize  the  important  formulcB  used,  and  render  their  meaning  clear.  Their  precise  practical 
value  cannot  be  appreciated  until  the  student  has  made  further  progress] 


IMPLICIT   FUNCTIONS. 


du 


102,  I*VOp. — Having  f  (x,  y)  =  0  =  u,  -r^  =  —  —-  ;   in  which 

— ,  and  —  are  the  partial  differential  coefficients  of  the  function  taken 
with  reference  to  x  andj  respectively. 

-r.  -n,  ^^x        1         rdvr\        du   ,   du  dy     ^  , 

Dem. — From  {99}  we  have  I  —  =  - — j_  — -  _£.     But  as  u  remains  constantly 

equal  to  0,  for  all  simultaneous  values  of  x  and  y,  when  both  x  and  y  have  taken 

on  contemporaneous  changes,  du  =  0.    Therefore  I  -—  =  0,  and  —  4-  — -  -^  =  0  : 
^  LdxJ  dx      dy  dx 

du 

.  dy  dx 

whence  -^  =  —  -r-.     q.  e.  d. 
dx  du 

dy 

ScH. — It  is  to  be  observed  that  though  I  y  J  =  0,  it  by  no  means  follows 

du  du        ^       _  ,  ,  „  .       „     , 

that  -y-,   or  --  =  0.      For   example,  y^  -\-  x^  —  r^  =  0  is  of    the  form 

f{x,  y)  ■=  0.  Now  if  y  changes  and  x  does  not,  the  function  changes  and  is 
no  longer  :=  0.  So  also  if  x  changes  and  y  does  not,  the  function  is  not  0. 
Bub  if  both  change  together  according  to  the  law  of  their  mutual  depend- 
ence, i.  e.  if  the  changes  are  what  we  have  called  contemporaneous,  the 
function  remains  equal  to  0 ;  and  its  total  differential  is  0.  [The  student 
can  observe  the  geometrical  signification  of  these  statements,  by  noticing 
that  y-  -\-  x'^  —  r2  r=  0  is  the  equation  of  a  circle,  and  that  the  function 
is  0  when  x  and  y  vary  together,  according  to  their  mutual  dependence  : 
but  when  one  varies  and  the  other  does  not,  the  function  varies  ;  i.  e.  the 
point  falls  out  of  the  circumference.  Illustrate  in  like  manner  from  y-  — 
2px  ^  0.] 

dy 
Ex.  1.  Given  x^  +  ^ax^y  —  ay^  =  0,  to  form  -r-  npon  the  principle 

0,00 

just  demonstrated. 

Solution.  —  Putting  m  =  0  =  cc^  -|-  '2>ax'^y  —  ay^,  we  have  dxU  =  4.x"dx  -|-  4:axydx, 
and  dyii  =  2ax^dy  —  Say^dy  ;   whence  we  have  the  partial  differential  coefficients 

du 

da;  '         ^  dy  dx  du  2ax^  —  Say* 


52  THE  DUTERENTIAL  CALCULUS. 

Ex.  2.  Given  ax^  +  ^'^V  —  ^V'  =  0,  to  form  — -  as  above. 

ax 

^      ,,    dy  Sax^  4-  Sar^v 

Result,  -f-  = ^. 

ao;  X-'  —  oay^ 

dy 
Ex.  3.  Given  y^  —  2axy  -{-  x^  —  b"^  =  0,  to  form  -^  as  above. 

ax 

dy        ay  —  x 

Result,  v-  = • 

ax       y  —  ax 

dxj 
Ex.  4.  Given  y^  —  Sy  +  ^  =  0,  to  form  -f-  as  above. 

dx 

Result  -^  == 


'  dx       3(1  —  y^y 


dy 
Ex.  5.  Given  x^  —  y"  =  0,  to  form  — ^  as  above. 

dx 

jtesuu,  ^  =  y'-^y'^'gy 

dx        x^  —  xy  log  X 

V         / = —  dv 

Ex.  6.  Given  a^   +  v  secfo:?/)  =  0,  to  form  ~  as  above. 

dx 


_,      ^,    dy               yV  BeQ.{xy)i?in(xy)  4- 20"  yx^~^\o^a 
Result,  -^  = '■ — .  —^ ^-^- — . 

^^  xV Beo,  {xy)  tan  (xy)  -\-  2a''''x^logalogx 


COMPOUND   FUNCTIOISS. 

103»  Bef. — A   Compound  Function  is  a  function  of  a 

function.  Thus,  if  u  =  f{y),  and  y  =  (p{x),  u  =  y"[^(^)],*  and  u 
is  said  to  be  a  compound  function  of  x.  This  relation  is  often  indi- 
cated by  saying  that  "  u  is  a  function  of  x  through  y,"  or  that  "  u 
is  indirectly  a  function  of  x  through  y." 

In  case  u  =  y(^,  ?/)'  ^^^  2/  ^^=  ^(^)j  "^^  ^^J  ^^^^  ^  ^^  directly  a 
function  of  or,  and  also  indirectly  through  y. 

104.  Frop.—If  u  =  f(y)  and  y^  cp{x),  ;^  =  ^  ^- 

Dem. — From  u  =  f{y),  "we  have  du  =  ;7-cZ2/-     ^^t  from  2/  ==  ^(aj)>  "we  have  dy  = 

dy 

dy  ^       _  ,         du  dy  ^  ^  du       du  dy 

—da;.     Hence  du  =  —  -r-d^,  and  ^-  =  —  ^-     Q-  e.  d. 

ax  dy  a.^•  dx      dy  dx 

ScH. — It  will  be  seen  that  this  is  only  a  particular  case  of  the  preceding  ; 
but  it  is  of  such  frequent  occurrence  that  it  is  thought  best  to  give  it  prom- 
inence. 

*  Read,  "u  eqiials  the/  function  of  tbe  (p  function  of  x." 


SUCCESSIVE  DIFFERENTIATION  OF  FUNCTIONS  OF  TWO  VARIABLES.      53 

Ex.  1.  Given  u  =  a^,  and  y  =  6',  to  form  ~  on  the  principle  just 
demonstrated. 

r,  <:^u  ,  .  dy        .   .     .  du        dudy  . 

Solution.     —    =  ay\o3,a,  and  -^  =  &=^log&.     .•.-—=   —  — -  =  a«'loga  X 
dy  °  dx  dx        dy  dx 

&*  log  6  =  ayh^  log  a  log  &,  or  a*  iF  log  a  log  h. 

du 
Ex.  2.  Given  u  ==  6t/-»,  and  y  =  ax^,to  form  —  as  above. 

CtJu 

du        ^    _ 
dx 

du 
Ex.  3.  Given  u  =  log  y,  and  y  ==  log  x  to  form  —  as  above. 

ax 

du       1     1  1 


dx       y    X       xlogx 

10S»  JPvop, — Having  given  ?i=^(z)  Sbndz=f(x,  y)  to  differen- 
tiate u  with  respect  to  x  and  y  without  previously  eliminating  z. 

Dem. — Since  w  is  a  function  of  x  and  y,  we  have 

Now  w  is  a  function  of  x  through,  z  (i.  e.  it  is  a  compound  function  of  a).     Hence 

—-  =  —-—•  and  for  a  like  reason  -—==--  -^  (lOd).     Therefore,  substituting, 
dx       dz  dx  dy      dz  dy 

du  dz  ^      ,   du  dz  , 
du  =  —  --dx  4-  -r-  -T-dy.     q.  e.  d. 
dz  dx  dz  dy 

ScH. — The  student  should  not  fail  to  observe  that  all  the  truths  developed 
in  this  section  are  but  deductions  from  the  proposition  that  the  total  differ- 
ential of  a  function  of  several  variables  equals  the  sum  of  the  partial  dif- 
ferentials. With  this  key  in  hand,  he  can  readily  unlock  the  mysteries  of 
the  whole  subject. 


■^♦» 


SUCTION   VL 

Successive  DifFerentiation  of  Functions  of  Two  Ikdependent 
Variables,  and  of  Implicit  Functions. 

106,  JPvop, — In  a  function  of  two  independent  variables,  both  va- 
riables may  be  considered  equicrescent ;  i.  e.,  their  differentials  may  be 
regarded  as  constant. 

IrjL. — This  proposition  is  an  axiom,  and  it  is  only  necessary  that  its  import  be 
clearly  understood.  Thus,  if  u  =f{x,  y)  and  x  and  y  are  independent,  any  change 
which  X  may  undergo  does  not  affect  y,  and  any  change  which  y  may  undergo 


54  THE  DIFFERENTIAL  CALCULUS. 

does  not  affect  x,  as  this  is  what  is  meant  by  their  being  independent.  We  may 
therefore  conceive  each  of  them  to  change  according  to  any  law  we  please  ;  and  it 
is  found  convenient  to  conceive  that  x  increases  by  equal  infinitesimal  increments, 
as  heretofore,  and  that  y  also  increases  by  equal  infinitesimal  increments.  Thus 
dx  and  dy  are  constants  ;  but  it  does  not  follow  that  we  are  to  regard  dy  =  dx. 
In  fact  this  would  be  to  estabUsh  a  relation  between  x  and  y,  and  hence  would  be 
contrary  to  the  hypothesis 

107,  Def. — When  u  =  f{x,  y),  and  x  and  y  are  independent  of 
each  other,  dj,u  and  dyU  are,  in  general,  functions  of  x  and  y,  and 
hence  may  be  differentiated  with  respect  to  either  variable,  thus  ob- 
taining a  class  of  Second  I^artial  Differentials,  In  like 
manner  these  second  partial  differentials  are  in  general  functions  of 
X  and  y,  and  may  be  differentiated  with  reference  to  either,  giving 
rise  to  Third  I*artial  Differentials  ;  etc.,  etc. 

lOS,  JS^otatton. — Havmg  u  =  f\^x,  y),  -i—^'  TIT  ^  ^  ^^ 
dydx^  and  -^—dy^  are  the  symbols  for  the  second  partial  differen- 


dydx         ^  dy^ 

d^u 
tials.      The   third   partial  differentials   are   indicated   thus,    -^dx^, 

d^u    ,     ,  d'^u    ,    ,         d'^u    ^    ,  d^u    ^     ^  ^    dHi 

-dx^dy   or    --t — dydx%  dxdy^   or    ■j----dy'dx,  and    -^dy\ 


dx"-dy  dydx^  ^      '   dxdy^  dy'^dx  '  dy^ 

In  each  case  the  form  of  the  numerator  indicates  the  number  of  dif- 
ferentiations, and  the  denominator  the  variable  or  variables  with 
which  the  successive  differentiations  have  been  made,  and  the  order. 

Thus  ——-diMx  signifies  that  u  =  f{x,  y)  has  been  differentiated 
dymx 

three  times  in  succession,  twice  with  reference  to  y  and  once  with  re- 
ference  to  x,  and  in  this  order.  So,  in  general,  ^  dx^-^dy""  sig- 
nifies that  u  =f{x,  y)  has  been  differentiated  m  —  n  times  in  succes- 
sion with  reference  to  x,  and  then  n  times  with  reference  to  y. 

Ex.  1.  Given  u  =  x'^y'^  to  form  the  several  successive  partial  dif- 
ferentials. 

dii  du  d'-u 

Remits,  —dx    =    2y^xdx,    —dy    =    2x-'ydy ;      t^/^^   =    '^y^dx'', 

-dxdy    =^    4:xydxdy ;      -f-dy^    =     ^xmy"~ ;      t-«^^    =    0, 


dxdy       ^  ^       ^  '     dy'  ^  dx' 

d^u    ^     ^  .77        d^u    .    ,  A    J  J  .      d^^j  ,         A 

j^d^dy  =  iydx'dy,  j^^dxdy^  =  ^^dxdy- ;    -dy'  =  0 ; 

dhi 
-^-^-dx'dy^  r=  ^dx^dy\  etc. 


SUCCESSIVE  DIFFERENTIATION  OF  FUNCTIONS  OF  TWO  VARIABLES.     55 
Sug's. — Having  -^-dx  =  2y^xdx,  to  differentiate  it  with,  respect  to  jc,  we  notice 

in  the  first  member  that  it  will  give • dx  ;  the  dx  being  written  in  the  de- 

nominator,  and  as  a  factor  to  designate  with  respect  to  which  variable  the  differen- 
tiation is  made,  and  also  in  accordance  with  the  principle  that  the  differential 
coefficient  multiplied  by  the  differential  of  the  variable,  is  the  differential  of  the 
function.  Now,  the  differential  coefficient  being  the  differential  of  the  function 
divided  by  the  differential  of  the  variable,  we  have  for  the  differential  coefficient 

du  K^^"")  Kd'x^'d 

of  --dx  taken  with  respect  to  x   -—.  •  hence  the  differential  is  -dx, 

dx  ax  dx 

Finally,  observing  that  dx  is  constant,  this  becomes         ^   da;,  or  -jT^dx^.     In  like 

manner  the  student  should  analyze  the  other  processes.  It  is  of  the  utmost  impor- 
tance that  he  fully  comprehend  the  reasons  for  these  processes ;  if  he  do  not  he  will 
become  hopelessly  entangled  in  the  subsequent  operations. 

Ex.  2.  Produce  the  successive  partial  differentials  of  u  =  {x  -{■  2/)"* 

with  respect  to  x  ;  also  with  respect  to  y. 

du 
Results,  d^u  or  -^dx  =  m(^  +  y)"'~^dx, 

—dx^,  or  dj,d^u  =  m{m  —  l)(.r  +  y)"'~^dx^, 

dHi 

-r-d^^  o^  d^d^d^u  ^=  m{m  —  l)(m  —  2)(^  +  yY~'^dx^,  etc. 

ScH. — ^When  u  =/{x,  y),  the  following  forms  are  called  Partial  DiffeV" 

..    ,  ^     ^   >      ^       d^u    d-u      d-u     d?u      d^u        d^u  d"'u 

enttal  Coefficients :   -5—,   — -,  - — --,  — -,   - — — ,  -—7—, ; — -, 

dx^    dy^    dydx    dx^    dx^dy    dxdy^  dx^^^'dy" 

etc. 

Ex.  3.  Form  the  successive  partial  differential  coefficients  of  w  = 

sin  (a;  +2/)  with  respect  to  y. 

^      ^^    du  /      ,     s    d^u  .    ,  ^    d^u 

Besidts,  —==  Gos{x  -^  y),  —  =  —sm{x-j-y),  —  =  —cos{x-\-y), 

d-^u         .     .  .    d°u  , 

—  =  sm  {x  +  y),  — -  =  cos(^  +  y),  etc. 

Ex.  4.  Form  the  successive  partial  differential  coefficients  of  w  = 
cos(^  . —  y)  with  respect  to  x. 

^      J,     du  d-'u  ,  .    d^u        .    ,  . 

EesultSy  -5-  =  — smix — y),  — -  = — cosix — v),  -7— ==  sm(a:— v), 

etc. 

Ex,  5.  Form  the  successive  partial  differential  coefficients  of  w  = 


56  THE  DIFFERENTIAL   CALCULUS. 

log  {x  +  y)  with  respect  to  x,  and  also  with  respect  to  y  in  the  com- 
mon system  of  logarithms. 

J,     du  m        d^u  m  d^u  2m(x  -\-  y)  

dx         X  -\-  y    dx^  {x  +  y)'    dx^            [x  +  ?/)4 

,  etc.     The  partial  differential  coefficients  with  respect 


to  2/  are  altogether  similar, 


100.  JPfop, — Jf  u  =  f(x,  y),  in  which  x  and  j  are  independent, 
and  several  differentiations  he  performed^  m  with  reference  to  one  variable 
and  n  with  reference  to  the  other,  the  result  is  the  same  whatever  the  order 
of  the  ope7^ations. 

^  -  ,     ,^     -,        .,    ,    d'^u  ,   ,  d'^u  ,   , 

Dem.  —  1st.,  To  snow  that  ■        dxdy  =  -r—r-dydx. 

acco/y  aycix 

dn 

—dx  =f{x  -j-  dx,  y)  —f{x,  y),  and 
ax 

d-u 

-dxdy  =  f{x  -\-  dx,  y  -\-  dy)  —  f{x,  y  -\-  dy)  —  [/(«  +  dx,  y)  —f{x,  i/)]   = 


dxdy 

f{x  -\-  dx,y  -\-  dy)  —f{x,  y  +  dy)  —f{x  +  dx,  y)  +  f{x,  y). 

Again,  -^-dy  =  f{x,  y  +  dy)  —fix,  y)  and 

d'^u 

dydx  =  f{x  .\.  dx,y  -\-  dy)  —  f{x  +  dx,  y)  —  \_f{x,  y  +  dy)  —  f{x,  y)1  = 


dydx 

fix  -{.  dx,y  +  dy)  —fix  +  dx,  y)  —fix,  y  -\-  dy)  -\-  fix,  y). 

These  two  results  being    identical,  we  have  -——dxdy  =  dydx,  (1). 

2nd.,   To  show  that  - — -  dx'^dy  =   ,    ,    dydx^. 
dx^dy       ^        dydx'^  ^ 

d^u  d/hi 

For  convenience  of  notation  put  dxdyU  for  y-r  dxdy,  and  dyd^u  for  ^— ^  dydx. 

Then,  as  before  shown  dydxU  =  dxdyU  =  fix,  y),  and  hence  may  be  differentiated 

with  reference  to  x  or  y.     Differentiating  with  reference  to  x,  we  have,  dxdydxU  = 

d'^u 

dsdxdyU*    But  by  (1)  dxdyidxU)  =  dydxidxu)  or  dydxdxU.     Whence  jiydx^  = 

In  Hke  manner  we  may  proceed  to  any  extent  desired. 

d^u  d^i 

[Let  the  student  show  that  t; — ir-dx^dy^  =  -; '    ,    dy-dx^"] 
*•  dx^dy^       ^        dy^dx^  ■' 

Ex.  1.   Given  u  ==  xlogy  in  which  x  and  y  are  independent,  to 

form  the  several  second  and  third  partial  differentials,  and  to  show 

that  dydxU  =  dxdyU,  and  also  that  dydxdxU  =  dxdydxU  =  dxdxdyU.* 

-n      T,     d^u  ^       . .        ,  7    ,  ^      dHi  ^    ^     , .        -,  ,    X         d^dy 

Results,  -r—dx^  (i.e.  dxdxU)  =  0,    - — —dxdy  {i.e.  dxdja)  =  , 

'  dx"^    ^  ""  ""  ^  '   dxdy       ^  ^  '   ^  y 

*  These  opieratione  are  sometimes  indicated  thus  :  d„u=  d^u,  d.„ji=  d^,„u=  d_„u,  etc. 

yx  xy  yx*  xjf*  dcx|f 


SUCCESSIVE  DIFFERENTIATION  OF  FUNCTIONS  OF  TWO  VARIABLES.    57 

T    d"u  dydx    dHc ,  xdu'^      d^u    ,    , 

and  - — —dydx  also  =   -^ ,  -7—dy^  = '—,  - — -—dxdy'^  == 

dydx  ^  y       dy'i  ^  y      dxdy^       ^ 

dxdi/^        -     d^u 

'—,  and  -; — 7—  =  0. 

y"  dydx^ 

Ex.  2.  Given  u  =  x^y  +  ay"^,  to  form  the  second,  third,  and  fourth 
partial  differential  coefficients,  and  show  the  convertibility  of  the  in- 
dependent differentiations. 

_,      -,     du      ..         du  ,  _.        d'^u       ^       d^u      _       d^u 

Hesults,  -rr-  =  ox'^y,  — -  =  ^3  _l  2ay:  -—  =  bxy,  — —  =  2a,  -; — —  =  dx^ 
dx  ^    dy  ^    dx'^  ^   dy^  dxdy 

_    d'^u         ^       ,         d'^u        ^     d^u       ^     d^u         ^  d-^u 

and  -; — —  =  Sx"  also  ;  - —  =  by,  — —  ==  0,  - — 7-  =  6x  =  — — -— , 
dydx  dx^  dy^  dxmy  dyax^ 

d^u  ^         d^u        d*u        ^     ^ 

0  =  3-—^-  ;    3—  =  0,  etc.,  etc. 


dxdy'^  dy-'dx      dx'^ 

Ex.  3  to  7.  As  above  form  and  compare  the  successive  partial  dif- 

^2   yi  J^ 

ferential  coefficients  of  the  followine^  :    u  = :   u  =  tan~^-  : 

u  =  sin  X  cos  y  ;  u  =  x^ ;  and  u  =  {x  -\-  y)'*. 


110,  JPfob, — To  form  the   successive   differentials  of  a  function 
of  two  independent  variables. 

Dem.  —Let  u  =f{x,  y),  in  which  x  and  y  are  independent  variables.     The  total 
differential  being  equal  to  the  sum  of  the  partials,  we  have 

Now,  remembering  that  as  x  and  y  are  independent  and  hence  may  be  treated  as 

equicrescent,  dx  and  dy  may  be  considered  constant,  and  remembering  also  that 

du  du 

— ,  and  —  are,  in  general,  functions  of  x  and  y  {107),  we  proceed  to  differentiate 

€vvO         ^y 

(1)  again.     Thus 

and  \dy)'='d^x^''-^  d^^^y^        which  substituted  give 

d'^u  o^^w  J  J     ,    d;^'^  J    1     ,  d^u.  ^      d^u^  „  ,  „  d^u  ^    ^     ,  d^u , 

d-'u  =  7— <^^2  _^  ^^ydx  -f-  ——dxdy  4-  5— dys  =  -r-,dx^  4-^-—rdxdy  -\ dy^, 

dx-^  dxdy  dydx      ^   ^   dy^  ^        dx'^        '     dxdy  dy^  ^ 

dht  .J  dH  J   .    ^  -  -  ^^ 

'^^^^  Sy^^'^y  =  dyd^^y^""  ^^^^^' 

Again,  differentiating  this  second  differential,  we  have 

^  _  du 

*  To  perform  this  operation  observe  that  —  is  treated  as  a  function  of  x  and  y,  and  hence  its 

dx 

total    differential  is  equal  to  the  sum  of  its  partial  differentials.      The  partial  differential  with 

respect  to  aj  is  _2fdx,  and  with  reference  to  2/.  — —dy. 
(ja;2  dxdy 


58  THE  DIFFERENTIAL  CALCULUS. 


<^=o-+-o-^+<s>^-> 


a(  --. — ;-  )  =  -; — T-dx  +    ; — r-dy, 
\dxdy/       dx^dy  dxdy^ 

and  cZ(  -—  )  =  dx  A-  -^—dy.  Substituting,  we  have 

Kdy'^J       dy^dx        '    dy^  ^  ^ 

d^w ,       ,       d^u   ,     ^      ,    ^  d'^u   ,     ^      ,    ^  dht    ,    ,       ,      d^u   ,     ^      ,    d% , 
=  -v-cZx3  _L  -^ — -dx^dy  4-  2- — -dx-dy  4-  2  ,    ,    dxdy'^  4-  ——-dy^dx  4-  -—dy^ 
dx^        ^  dxHy        '^  ^    dx;^dy       "^    '     dxdy'^       "^    ^  dy^dx  ^       ^  dy^  ^ 

^^^    7      ,  .  O       <^^W         7      »    7  ,  r,       ^^^*         7      „    7  ,  <^^W    7      , 

=  -r;-dx^  4-  3- — -dx^dy  4-  3- — -dy^dx  4-  —-dy\ 
dx^       ^    dxHy        -^  ^    dyHx  -^       ^  dy^  ^  . 

d^u    ,  „,  d-u       _   ,   _  dhi   ,   ,  „     ^ 

since  -; — --dx-dy  =  ^— ; — —  dxdydx  =  --^ — dydx%  etc. 
dx^dy  dxdydx        ^  dydx^  ^ 

In  like  manner  we  may  proceed  to  differentiate  as  often  as  desired. 


ScH. — A  little  observation  will  enable  the  student  to  write  out  any  re- 
quired differential  of  it  =  /{x,  y)  by  analogy  from  the  above.  He  only- 
needs  to  notice  that  every  distinct  form  of  the  partial  differential  of  the 
required  order  is  involved,  and  making  x  the  leading  letter  insert  the  coef- 

ficients  as  in  the  binomial  formula.     Thus  d^u  =  —  'dx''>  +  5 dx^dy  -j- 

dx^  dx^dy 

^^   d'u     ,  ,  ,  „    ,    ^„    d^u     ,  „  ,  „    ,    _  d'm    ^    ,       ,    dm  ,  . 

10- — —dx^dy'^  +  10 dx'^dy^  +  5 dxdy*  4 dy: 

dbfldy^        ^  dx'^-dy^        ^  dxdy^       ^         dy'^  ^ 


111,  ^TOh, — To  form  the  successive  differential  coefficients  of  an 
implicit  f  miction  of  a  single  variable. 

Solution. — ^Let  u  =f{x,  y)  =  0,  in  which  y  is  an  implicit  function.     We  are  to 

.        d%  d^y     ^ 
form  -7^,  — ^,  etc. 
dx2'  dic3' 

du 

dii           dx-  ' 

First  we  have  -^  = by  {102).     (1). 

dy 

The  form  for  differentiating  this  is 

/du\du         /du\du 

d^y  \dx/7ly         \dy)dx    .    ,       .        „  ,  ,      •       ^    ^. 

•T-,  =  —  ,     „     •         —  dx,  since  the  second  member  is  a  fraction. 

\dy) 

To  perform  the  operations  thus  indicated  we  have  to  remember  that  -r-  and  tt- 

dx  dy 

are  functions  of  x  and  y. 


Hence  d(  —  )  =  --dx  4-  -r—-dy* 
\dx/       (Zx2      ^  dxdv^ 


SUCCESSIVE  DIFFERENTIATION  OF  FUNCTIONS  OF  TWO  VARIABLES.    59' 

and  d(  ~~)  =  - — —dx  4-  -r-dy.     Dividing  these  values  by  dx  and  substituting:, 
\dy/        dydx  dy-  ° 

du/d-u         d"u    dy\       du/  d-u         d'u  dy\ 

d^j  diKdx'^       dx  dy  dx /       dxXdydx       dy^dxj 

we  have  -^  =  —    ^  7  .  >  ■ 

\dy) 

dii 
Finally,  substituting  in  this  the  value  of  —  as  given  in  (1)  we  have 


du  f  d-u  d'^u    dx  \         du  I    d^u 


I 


dy  \  dx^        dx  dy  du  |    dx  |  dy  dx 


dx^  /duy 

\dy) 
/du\^/d"u\  d^u    du  du  d"u    du  du        d^u/du\^ 

\dy/\dx-/         dxdy  dx  dy        dydx  dx  7ly        dyAclx) 

\d4/) 
d'U/duY        n  ^"^    ^^  ^^  _t    d^u/duY 
dxAdyJ  dxdy  dx  dy        dyAdx/ 


(- 
\d 


duy 


.dyJ 

In  like  manner  the  higher  coefficients  may  be  produced,  but  the  forms  are  too 
complicated  for  elementary  purposes. 

Ex.  1.  Form  the  first  and  second  differential  coefficients  of  y  as  a 
function  of  x,  when  y^  —  2axy  -f  x^  —  62  =  0. 

du 

dy  dx  —  2aw  4-2^7         ay  —  x      ^ 

Solution.     --  =  —  •—  = — —- =  .      For  convemence  of 

dx  du  2?/  —  2ax  y  —  ax 

dy 

notation  put  --  = =  p,  whence  p  is  a  function  of  x  and  y.     Hence  |  -  -    r= 

dx       y — ax      f  ^  ^  LdxJ 

dp*  dp*  dy 

dx  dy    dx' 

—  (y  —  ax)  4-  aiciy  —  ^)        ,  dp*  /ay  —  ^\    .    ^        ^''iy  —  ^'^^  —  («?/  —  ^) 


I    (99).      But    \f\   =  %    f   =   dri^^^)   -   dx 
X  LdxJ  dX'     dx  \y  —  ax/- 


and  f  =  d./'^^IH^)  ^  dy  =  2< 

dw  \y  —  ax/ 


Y2 


(y  —  ax)^  dy  \y  —  ax/    '  {y  —  ax) 

Reducing,  --  = -,  and  -~  = '-.     Substituting  these  values  and 

dx       {y  —  axy-i  dy  {y  —  ax)'-^ 

dv 
also  the  value  of  --  as  at  first  found,  we  have, 
dx 

d^y  _  (a^  —  Vy        (a^  —  l)a;       ay  —  x (a^  —  l)(y^  —  2axy  -f-  x'^) 

<^^      {y  —  ^^y^     (2/  —  ^^)^     y  —  «*  iy  —  «^)^ 

Ex.  2.  Form  the  first  and  second  differential  coefficients  of  ?/  as  a 
function  of  x,  when  y^  -\-  x^  —  r^  =  0. 

dy  X  d-^y r^ 

\'  dx  y'  dx^  T/3* 

*  Beiuembar  that  these  are  partial  differeutial  coafficidnts. 


60  THE  DIFFERENTIAL  CALCULUS. 

SuQ. — Be  particular  to  use  the  method  now  being  illustrated. 

Ex.  3.  Given  ^3  _}_  Saxy  -f-  i/3  =  0,  to  form  -j-,  and  —^,  by  the  method 

for  differentiating  implicit  functions. 

_       ,       di/  x'^  -\-  ay  d  y  ^a?xy 

Results,  -^  = 


dx  V'  -\'  ^•^'  ^'^■^        (Z/'  +  CLXf 

Ex.  4.  Given  y^  —  %jcy  -f  o^-  =  0,  to  form  the  first  and  second  differ- 
ential coefficients  of  ?/  as  a  function  of  x,  by  substituting  in  (1)  and 
''2)  of  the  preceding  demonstration. 

^     ,       du  ^      du       ^         „      d%  d^u  d^u  dy 


d^u/du\^       ^dH(    dii  du      d^u/du\~ 

2w                t^                  d'v               dxAdyJ        "dxdy  dx  dy       di/Adx/ 
,•/       —       ^  g^jj^    _^    __   — : 


0  .  (22/  —  2a;)2  —  2(—  2)(—  22/)(2y  —  2x^  +  2(—  2.y)2  ^  _  —mj:y  —  x)-\-8y'^  ^ 
(2?/  —  2iCj3  (2?/  ~  2a;/ 

y(y  —  2a;) 
(y  —  x)3  * 

.         dy        ^  d'^y  ... 

Ex.  5.  Given  cos  {x  +  y)  =  0,  to  form  -^,  and  —  by  substituting 

as  above. 

du  .         ,     ,  du  •    /     ,     N  <^'^^  ,     I     N    d^u 

—  cos  (a;  +  ?/),    -^  =  —  cos  (.r  +  2/).      Substituting,    --  =  —  1,    and    ~    = 

—  cos  (■'g + y)  sin-^(.r  -f  y^  4-  2  cos  (a;  -{- ;?/)  sin^.  x  +  y^  —  cos(a; -|-  ?/ ■  sin^  ;r  +  y^  _ 
'  —  tsm^^^x  4-2/; 

These  results  are  as  might  have  been   anticipated,  since  for  cos  ^.v  -^  y)  =  0, 
X  +  2/  =  90°  ;  hence  as  one  arc  (x)  increases,  the  other  {y)  decreases  at  the  same 

rate.     Therefore  -/  =  —  1,  and,  consequently,  t^  =  0. 
dx  a*'' 

Ex.  6.  Solve  Ex's  1 — 3  inclusive  by  substituting  in  the  general/orm- 
ulce  (1)  and  (2). 


DERITED   EQUATIONS. 

112.  From  w  =  0  =f{x,  y),  we  have 

du 

^  ==  _  —      (1) 
dx  du 

dy 


CHANGE   OF   THE   INDEPENDENT   VARIABLE.  61 

d'^u   dy\        du/  d"u        d^u  dy\ 
dxdii  dx)        dxSdiidx       dy-  dx^ 


du/d^u        d^u   dy\        du/  d-u        d^u  dy 

^  d'^y  dy\dx^       dxdy  dx)       dxKdiidx       dy  ^^ 

and— -  = ' T ,     (2), 

dx-^  /du^  ^   " 

\d^) 


-r,  ^x  -.  du  dy        du         ^       /-,  s       ,  •  ■,    .        ,,    -,    rw^^ 

From  (1),  we  nave   j~t^+3~^=0-     (J-O'  which  is  called  TJie 

First  Derived  JEqiiation,  or  The  Differential  Equa- 
tion of  the  First  Order, 

From  (2)  we  obtain 

,  du/d^u        d^u   dy\        du/  d^u        d^u  dy\ 

du  d^y  dy\dx^      dxdy  dx/        dxxdydx       dy^  dx/ 

dy  dx^  du 

dy 
du 
d'^u         d^u  dy      dx/  d'^u        d^u  dy\ 
dx^       dxdy  dx       du\dydx       dy^  dx/ 

Ty 
dHi         d-u   dy       dy/  d^u        d^u  dy\ 
dx^       dxdy  dx       dxXdydx       dy^  dx/ 
dHL  d'^u   dy       d/Hi/dyy 

dx^          dxdy  dx       dy^\dx/  ' 
Whence,  transposing,  we  have, 

du  d^y  d^u   dy       dHi./dy\"       d'^u 

dy  dx^         dxdy  dx       dy'^dx)     '   dx^         '     ^  ^^' 
which  is  called  The  Second  Derived  Fquation  or  The 
Differential  Fquation  of  the  Second  Order, 

In  a  similar  manner  the  Third  Derived  Equation  is  found  to  be 
du  d^y  f   dm         d^u  dy  ]  d^y        dH(./dy\^  d^u  /dy\^ 

dy  dx^  I  dxdy        dy^  dx )  dx^        dyAdx/  dxdy^dx/ 

^  d^u    dy        dHi 
ax"ay  dx        dx'^ 


SECTION   YIL 
Change  of  the  Independent  Variable. 

113,  In  considering  functions  of  a  single  variable,  as  ?/  =  f{x), 
the  hypothesis  which  we  usually  make  that  x  is  equicrescent,  and 
hence  that  dx  is  constant,  gives  to  all  the  differentials  and  differential 
coefficients  of  the  function  after  the  first,  a  different  form  from  what 
they  would  have  had  if  such  hypothesis  had  not  been  made.     Thus 


&2  THE   DIFFERENTIAL     CALCULUS. 

^/dy\       d^     ,           .          .              ,■         d^ydx  —  d'^xdy 
d{-r-\-=  —-,  when  x  is  equicrescent,  but —— '-,  when  neither 

•variable  is  regarded  as  equicrescent  (i.  e.  when  dy  and  dx  are  both 
treated  as  variable).  In  the  course  of  a  discussion  it  sometimes  be- 
comes important  to  change  the  conception  and  regard  y  as  the  equi- 
crescent, or  independent  variable,  and  x  as  the  function.  Or  it  may 
be  desirable  to  introduce  a  new  variable  of  which  ^  is  a  function,  and 
make  it  the  equicrescent  variable. 

Either  of  these  changes  can  be  readily  effected  by  first  giving  to 
the  expression  under  consideration  the  form  which  it  would  have  had 
if  neither  variable  had  been  treated  as  equicrescent.  Then,  to  make 
y  equicrescent,  remember  that  all  its  differentials  above  the  first  are  0, 
and  drop  out  the  terms  affected  by  them.  To  introduce  a  new  inde- 
pendent equicrescent  variable,  as  6,  of  which  j:  is  a  function,  simply 
substitute  in  the  general  form  in  which  neither  x  nor  y  is  equicres- 
cent, the  values  of  x,  dx,  d-x,  etc.,  in  terms  of  the  new  equicrescent 
variable  0, 

dy   d'Y   d^y 
114z.  JProh, — To  find  the  forms  which  ^,  — -,  —-,  etc.,  take  when 
-'  ^  dx   dx^   dx^ 

neither  variable  is  considered  equici^escent. 

Dem. — Since  the  hypothesis  of  au  equicrescent  variable  has  not  modified  the 

form  of  --,  in  it  a;  or  v  may  be  considered  equicrescent,  or  neither,  at  pleasure. 
dx  " 

Again  —   =  — —— .     Now  differentiating  the  latter  without  regarding  dx  as 
dx'  dx 

/  dy\        d^y  dx  —  d'^x  dy 


<l) 


\dx/  dx^  d^ydx  —  d^xdy      ,     ,   .    . ,       ^ 

constant  we  have  -X—-  =  ^ =  -' rz ^'  which  is  therefore 

dx  dx  dx^ 

the  form  which  the  second  differential  coefficient  takes  when  neither  variable  is 

equicrescent. 

\/c^^y\  j^d-^ydx   —    d-^xdy\ 

d^y  \d^^)  ""{  dx^  ~) 

Once    more,    -r-^    =    -    ,  =    • -= 

dx^  dx  dx 

d3ydx*  -f  d^yd^xdx^  —  d^x  dy  dx^  —  d'^x  d^y  dx^  —  3(d3y  dx  —  d'^x  dy)dx^  d^x  _ 

jd^ydx  -  d^xdy)dx  -  Sid^ydx  -  d^xdy)d^x^  ^^.^^  ^^^^^^  .^  ^^^  ^^^  ^^^^^^^  ^^ 

dx^ 
the  third  differential  coefficient  when  neither  variable  is  equicrescent. 

Ex.  1.  Transform  jt—  4-  (-^)  —  -^  =  0  in  which  x  is  equicres- 
dx-^       \dx^         dx 

cent,  into  its  equivalent  when  y  is  equicrescent. 

^          d^y       d^y  dx  —  d^x  dy 
Solution. — When  y  is  equicrescent  d^y  =  0,  hence  —  =  -^-^ 


CHANGE  OP  THE  INDEPENDENT  VARIABLE.  63 

d'^xdy     ^  .   ^.,   ^.  -  d'^xdy   ,   dy^        dy        „      ^.  .,. 

—  ~lb^'     S^^s*^*'^*^^^'  ^^  ^^^e  -  ^-^  +  ^3  —  dx        ^-     Dividing  by  dy3 

and  multiplying  by  dx,^  to  give  tbe  differential  of  the  independent  variable  its 

d^x       /dx\2 
proper  position,  and  changing  signs,  we  obtain,  x- \-  \-Jr)  —  1=0. 

3. 

Ex.  2.  Transform  {dy^  +  dx^Y.  -\-  adxd^y  =  0,  in  which  x  is  equi- 
crescent,  into  its  equivalent  when  y  is  equicrescent. 

Result,  (l  +  ^')*  —  at-  =  0. 
\         dy^/  dyi 

Ex.  3.  Transform  -p-  —  —^ —  -^  +  -— ^ —  =  0,  in  which  x  is 
dx^        1  —  x^  dx        1  —  a;2  ' 

equicrescent,  into  its  equivalent  in  terms  of  0  as  the  equicrescent  va- 
riable, when  X  ==  cos  6. 


SoiiUTioN. — Introducing  the  general  form  of  the  second  differential  coefficient, 

^,       .  j^.       ■,  d:^y  dx  —  d^ic  dy  x      dy   ,        v 

the  given  equation  becomes,  — ^ r-— ~  -i ^ —  =  0. 

^  ^  dx»  1  —  x2  dx  ^  1  —  x^ 

Now,  from  a;  =  cos  6,  dx  =^  —  sin  0  dO,  dKx  =  —  cos  6  dB%  and  1  —  x^  =  sin^  9. 

a  V  t.-i.  4.-       J.-U  1  I.         — d^y sin QdQ-\- COS SdO'-^dy   ,     cosQ       dy 

Substituting  these  values,  we  have, — ±  -4 L 

—  sm3  e  d93  ^  sin2  6  sin  ddS^ 

v  "  d^v 

—  0,  or  reducing,  -^  -{.  y  =,  0. 


sin-^0         '  ^'  de^ 

Ex.  4.  Transform  R  == — ,  in  which  x  is  equicrescent,  into 

dx^ 
its  equivalent  in  terms  of  the  variables  r  and  6,  6  being  the  equicres- 
cent variable,  when  y  =  r  sin  0,  and  x  =  r  cos  (9. 

Sug's. — The  formula  in  its  more  general  form,  in  which  neither  variable  is  re- 
garded as  equicrescent  is 

(l  +  ^^f 

_     \    ^  dx^J       _     (dx2  4-  dy2)2 


d^y  dx  —  d^x  dy        d^y  dx  —  d~x  dy 
dx^ 
From  y  =  rsinS,  we  have,  dy  =  sinQdr  -\-  rcosQdB,  and  d^y  =  sin  9  d^r  -f- 
2  cos  9  dQ  dr  —  r  sin  0  d02.     From  x  =  r  cos  9,  we  have,  dx  =  cos  0  dr  —  r  sin  9  d0, 
and  d-cc  =  cos  9  d^r  —  2  sin  QdQ  dr  —  r  cos  0  d92.     Substituting  these  values  and 
reducing,  we  have, 

/dr^    .       M 

„       im  +  '-V 


_dr2  d^r    , 

d02         d02  ^ 


J[J[5.  The  method  just  given  is  sufficient  to  resolve  all  cases  of 


64  THE   DIFFERENTIAL   CALCULUS. 

change  of  the  equicrescent  variable,  but  general  formulas  are  some-  | 

times  convenient.     We  proceed  to  deduce  the  most  important. 

%1G»  JPvop, — In  operations  where  j  has  been  treated  as  a  function 
of  the  equicrescent  variable  x,  to  change  the  conception  so  that  x  shall  be  a 

dy     1 

function  of  the  equicrescent  variable  y,  we  substitute  for  -^,  —  and  for 

d^x 
d2y  dy2 
dx^'        dx3* 


Dem. — As  tlie  hypothesis  of  the  equicrescent  variable  does  not  affect  the  first 

differential,  we  have  the  identical  relation  -^  =  — . 

ax      dx 

dy 

.     .      .^      ..,            •  ,1     .          .            J.         1          d^y       d^ydx  —  d^xdy 
Again,  if  neither  vanable  is  eqmcrescent  we  have    -r—  = — . 

d^ 

•  ,        ,        ,  ^     ,  d^y  d'^xdy  du^ 

Now,  making  y  equicrescent  makes  d^y  =  0  ;  hence  ^  =  ■ — -^  =  —  rr-. 

**  ^  da^  dx^  dx^ 

Q.  E.  D. 

ScH. — In  a  similar  manner  the  corresponding  forms  for  the  higher  coef- 
ficients can  be  deduced ;  but  they  are  not  often  required. 


lit*  IPvop* — In  expressions  where  y  has  been  treated  as  a  function 

of  the  equicrescent  variable  x,  to  change  the  expression  so  that  y  shall  be  a 

function  of  some  new  equicrescent  variable  as  d,  having  given  x=  <p{^), 

dy  d-y  dx       d^x  dy 

^     dy      d^        ,  ^     d2y      d^  d^  ~  d^2  d^ 

we  substitute  for  ~^,     rr-,  and  for  - — ,     = — — — 

^      dx      dx         -^      dx2'  dx3 

d^  d^ 

dv       dy  dx 
Dem. — Since  y  =  f{x)  and  x  =  <p{B),  we  have  {104),  -—  =  ~  %o  ;  whence 

ao        dx  u\j 

dy 

dy  _  dS 

dx       dx' 

Te 

d-y 
Again,  the  general  value  of  -r-^  (when  neither  variable  is  treated  as  equi- 

crescent)  is 

d^y  d'^y  dx  —  d'^x  dy 

dx^  dx'^ 

dx 
Now  from  x  =  <p'6)  we  have  dx=-jdO.     Differentiating  this,  remembering  that 


CHANGE  OF  THE  INDEPENDENT  VABIABLE.  65 

dS  is  constant,  we  have,  d^x  =  -^^dB\      Substituting    these  values  of   dx   and 

d^x,    in    the     general    value    above,    we  obtain     — -    =   — ^— ^    = 

^  da;2  dx^ 

dAe  -  ^de^  dv      ^lM^-^^I 


^-dS^  ^ 

dQ^  d93 


Q.  E.  D. 


ScH. — To  apply  these  formulce  in  practice,  we  make  the  requisite  substi- 

d]/  (Py  dx       d'^x  dy 

i.  X-  £^    o      <^y        ^dB'~dB~~d^^'dB  .      d^-y  .     ,.        . 

tutions  of  —  for  — ,  and  — — ^ for  ~-  m  the  given  expression, 

ctcc         (xoo  doc  (XtC 

de  7m 

dx  d'^x 

and  then  finding  the  value  of  —  and  of  —   from  the  relation  x  =  ©(0), 
^  dd  dS'i  ^  ' 

substitute  these  values,  and  have  an  expression  in  terms  of  —  and  — -,  as 

^  dS  dQ^ 

required. 

d^y  y 

Ex.  1.  Given  3 —  =  -j——f — tt,  i^i  which  x  is  the  equicrescent  (in- 
dx^        {e''  +  e-^y  ^  ^ 

dependent)  variable,  to  transform  so  that  t  shall  be  the  equicrescent 

variable,  knowing  that  x  =  log  —  . 

Vl  —  t^ 

d^y 
Sug's. — Substituting  the  value  of  ^  requisite  for  this  transformation,  we  have 

d^y  dx      d^x  dy 

W  dt        m  dt  y 


ote3  (e^  +  e-=^)2 

d^ 

_                 -            t                 ,^.dx             1              ,  d^a;              1—3^2 
From  X  =  log  —  ,  we  obtain  — -  =  — — ,  and  --—  =  —  -— --.     In- 

troducing  these  values,  the  expression  becomes  -v^ 1- = 

^  '  F  ^^2  ^^-^  —  P)^  t\l  —  P'f  di 

(ex-t^e-^2  t\l-PY     ™'  ''^'''^*  readily  reduces  to  {t-pf^+a-m^  =  iy, 

by  observing  that  e^  =  —  .  and  e-*  =  ;   whence   (e^  4-  e-^)2  == 

1 


P{1  —  P) 

d^ii       1  dy 
Ex.  2.  Transform  ^^  +  -  -^  +  y  =  0,  into  a  form  in  which  t  is 

the  equicrescent  (independent)  variable,  knowing  that  x'^  =  U. 

» 

_,      -     dt^y       dy 
Result,  ^-il  +  -J  4-  y  =  0. 
dt^       dt 


6Q  THE  DIFFEBENTIAL  CALCULUS. 

ScH. — We  observe  from  the  foregoing  that  a  change  of  the  equicrescent 
variable  may  greatly  simplify  the  expression. 

I    Ex.  3.  Making  x  =  cos  t,  and  t  the  equicrescent  variable,  show  that 

,^  .d'-^y         dy       ^  .  d-y       ^ 

(1  — ^^)t^  —  x-r  =  0,  becomes  -p-  =  0. 


118,  I^voh, — Having  u  =  f(x,  y),  where  x  =  <p{r,  6)  and  y  = 
^lif}  ^))  io  express  the  partial  differential  coefficients  —  and  —  in  terms 

of  r,  0  and  the  partial  differential  coefficients  — ,  ana  — . 

Solution. — Since  w  is  a  function  of  .^•  and  y,  each  of  which  is  a  function  of  r, 
we  have 

du        du  dx        du  dy 


And  in  like  manner 


dr        dx  dr        dy  dr 


du        du  dx        du  dy 

d6  "^  dx  dQ  ^  dy  dQ'     ^  '' 


._ -a  ^^y   ""-  ""^-a  ^"^ 

dx' 

du  dy 

du  dy 

du  dx        du  dx 

du        dr  dQ 
dx        dx  dy 

dQ  dr        .  .  .,  ,  du 
dy  dx                          dy 

drd^~~dQdr 
dx  dy        dy  dx' 

dr  dQ 

dr  dQ 

dr  dQ        dr  dQ 

110m  CoE. — y  X  =  r  cos  6,  and  y  =  r  sin  6^  the  above  formulce  he- 

du  ^  du        sin  ^  du        ,  du         .       du       cos  0  du 

come  — -  =  cos  0 -— ,  and  -—  =  sm  6- 1 -— . 

dx  dr  r     d^  dy  dr  r     d^ 

ScH.  1.^ — It  will  be  observed  that  the  relations  x  =  r  cos  Q  and  y  =  ?'  sin  Q, 
are  the  common  formulce  for  passing  from  rectangular  to  polar  co-ordinates 
(Pakt  L,  128). 


GEIITERAL   SCHOLIUM. 

We  here  conclude  the  subject  of  the  Differential  Calculus,  having  de- 
veloped the  theory  as  fully  as  the  plan  of  our  course  requires.  In  the  next 
chapter  we  shall  give  a  few  applications  ;  but  its  transcendent  efficiency  is 
best  seen  in  the  General  Geometry  and  in  Physics. 


CHAPTER  n. 

APPLICATIONS  OF  THE  DIFFERENTIAL  CALCULUS. 


SECTION  L  . 

Development  of  Functions. 

120,  Def. — A  Function  is  said  to  be  Developed  when  the  in- 
dicated operations  are  performed  ;  or,  more  properly^  when  it  is 
transformed  into  an  equivalent  series  of  terms  following  some  gen- 
eral law. 

Ill's. — Division  afifords  a  method  of  developing  some  forms  of  functions.     Thus 

y  = ;  vehen  developed  by  division  becomes  y  =  1  ^  x  -{-  x-  -\-  x^  -\-,  etc. 

The  binomial  formula  (Complete  School  Algebba,  105)  is  a  formula  for  develop- 
ing a  binomial.  Thus  y  =  (^a  -{-  xY  when  developed  becomes  y  =  a^  -f-  ^a'^x  -{- 
10a3^2  _|_  lGa?ic3  _|_  5^x4  -j-  .'^3.  The  subject  is  one  of  great  importance  in  math- 
ematics. 

MACLAURHs'S   FORMULA. 

121,  r>EF. — Mciclaurin's  Formula  is  a  formula  for  devel- 
oping a  function  of  a  single  variable  in  terms  of  the  ascending 
powers  of  that  variable  and  finite  coefficients  which  depend  upon  the 
form  of  the  function  and  upon  its  constants. 

122,  JProh, —  To  produce  Maclaurin's  Formula. 

Solution. — Let  y  =  f{x)  be  the  function  to  be  developed.  It  is  proposed  to 
discover  the  law  of  the  development,  when  the  function  can  be  developed  in  the 
form 

y  =zf{x)  =  A  -{-  Bx  -\-  ac2  4-  Dx^  -|_  ^x^  -f ,  etc., 

in  which  A,  B,  C,  D,  etc. ,  are  independent  of  x  and  depend  upon  the  form  of  the 
function,  and  its  constants. 

Producing  the  successive  differential  coefScients,  remembering  that  A,  B,  C,  Z), 
etc.,  are  constant,  we  have, 

^^  =  ^  -f  2ac  +  3Z>.r2  -f  4.Ex^  -f ,  etc., 
dx  I  '        ' 

;^  =  2C  -f  2  .  SDcc  +  3  .  4^2  +,  etc., 

^  =  2  .  3Z>  4-  2  -  3  •  4^x  -f ,  etc., 

^  =  2  .  3  .4J5:4-,  etc. 
Now  as  the  coefficients  A,  B,  C,  B,  etc.,  are  independent  of  x,  they  are  the  same 


68  APPLICATIONS   OF  THE  DITFEBENTIAI.  CALCULUS. 

for  all  values  of  it,  and  if  we  can  find  what  they  should  be  for  any  one  value  of  x 
we  shall  have  their  values  in  all  cases.  Now,  if  a;  =  0  we  have  (?/)  =/(0)  =  A, 
the  expressions   (?/),  and  /(O)  signifying  the  value  of  the  function  when  cc  =  0. 

--  (I)  =  ^'  (S)  =  ''■  (S)  =  -  -•  (i)  =  -  3  .  .i.,  t.e  (  )  si,n. 
fying  in  each  case  the  value  of  the  particular  function  when  x  =  0.     Hence  we 

find  A  =  (y),  B  =  (^\  C=  C?^),-^,  D  =  (^)  ^.  E=('^)  -^-,  etc. 
^^^  \dxr  \dxVl-2  Xdx^J  1.2.3'  WW  1.2.3.4 

Substituting  these  values,  we  have 

+,  etc., 


•  2.3.4: 
which  is  the  formula  required. 

123,  ScH.  1. — The  student  should  become  perfectly  familiar  -with  this 
important  formula,  and  for  this  purpose  it  will  be  well  to  describe  it  thus  : 
Maclaurin's  Formula  develops  y  =  f[x)  into  a  series  of  terms,  the  first  of 
which  is  tlie  value  of  the  function  when  x  =  Q  \  the  second  is  the  first  dif- 
ferential coefficient  of  the  function,  x  being  made  0,  into  x ;  the  third,  the 

x^ 
second  differential  coefficient,  x  being  made  0,  into  '— ,  etc. 

A 

124:,  ScH.  2. — This  formula  may  also  be  written   y  =  f[x)  =  /(O)  + 

/i(0)»  AiO),  etc.,  signify  the  same  as  {y),  \-j),  (^)'  ®*^-'  respectively. 

Ex.  1.  To  develop  y  =  {a  -{-  x)\  by  Maclaurin's  Formula. 

dv  d^xi 

SoiiTTTioN. — Differentiating    successively,     we    have    -^  =  7(a  -\-  xY,    -^  = 

dx  dx^ 

6.7(«4-cc)o,  ^  =  5.6.7(a+a:)4,  ^  =  4t'5'6'7{a-\-x)^  ^  =  3-4.5. 6-7(a+a;)2, 

^  =  2-3.4.5.6.7(a  +  x),  —  =  1-2. 3-4. 5-6-7.     Here  the  differentiation  termi- 

nates.     Making  x  =  0,  we  have,  (y)  =  {a -\-  0)7  =  a^,  (j-J  =  l(a  -\-  0)c  =  7aG, 

(^)  =  C.7a^(*^)  =  5.0.7a.,  f^^)  =  4.5.6.7«^  ^  =  3.4.5.6.7a^  ^^  = 
\dx2/  VdcV  \di;4/  dx*  da;^ 

2-3.4.5.6. 7a,  and  TI^)  =  1.2-3.4.5.6-7. 

Substituting  in  the  formula,  we  obtain 

/y-i  rjfi  /p4 

J/  =  («r  -f  .'r)7  =    «7  ^  7«6a;  -|-  6  -  7a"-  +  5  .  6  .  Ta^^  -f  4.5.6.7a3^-^  + 

qiyi  q^Q  ^qI 

or,  reducing,  y  =  {a  -\-  x^  =  a''  -]~  la^x  +  21^^x2  -f  35a'»ic3  -)-  35«^.'C^  +  21«9a;^  -f- 
7ax^  4"  ^'^j  3,  result  identical  with  that  given  by  actual  multiplication,  or  by  the 
Binomial  Formula. 


DEVELOPMENT  OF  FUNCTIONS.  69 

Ex.  2.  To  deduce  the  Binomial  Formula  from  Maclaurin's  Formula. 

.  Solution. — Let  t/=  {a-\-x)'"\  in  which  m  is  either  integral  or  fractional,  positive 
or  negative.     Then  differentiating  successively,  and  taking  the  values  for  x  =  0, 

we    have    (2,)  =  a",    (|)    =    ma-'-K    (g)    =    m(m  -  1)«"-.    (g)    = 

m(m  — l)(m  — 2)a"»-3,  (j^J  =  w(m  —  l)(m  —  2)(m  —  3)a"— *  +,  etc.  Sub- 
stituting in  Maclaurin's  Formula,  we  obtain  y  =  (a  -{-  x)'"  =  a"'  -\-  mct^—^x  -f- 
m{m  —  l)a^-'  —  -\-m{m  —  l)(m  —  2)a"»-^-—  +  "K'^  —  l)(m  —  2)(?n  —  3)a™—' 


2   '      '           ''            '         2-3    '       '           '"           '"  •         li.3.4 

-f-,  etc.,  or  we  may  write  y  =  [a  -\-  x)™  =  a'"  +  w^"*^^  -f T~^~' — ^'^    '^'  + 

m(m  —  l)(m  —  2)        „       ,    m(m  —  l)(m  —  2)(m  —  3)        ,       ,       ,         ■,.,.,■, 
_^ ^^ iam-zx^  ^ !^ '\         ^ -^""-^x*  +,  etc.,  which  is  the 

Binomial  Formula. 

Ex.  3.  Develop  y  =  sin  x. 

S.C..     (,)  =  0,  (I)  =  1,   (g)  =  0,   g  =  -  1.  eu.    .-.  .  =  sin.  = 

x^       ,  x^  x'  ,        , 

X + ,  etc. 

1.2.3^1.2.3.4.5       1.2.3.4.5.6.7^* 

Ex.  4.   Develop  y  =  cos  x. 

X^  X*  x^ 

Result,  y  =  cosx  =  l-—-^:^-^-^^-^^^^^^  + 

;  etc. 


1-2  -3    4-5-6    7-8 


125,  ScH. — These  formulce  enable  us  to  compute  the  natural  sine  and 

cosine  of  any  arc  directly.     Thus,  to  obtain  the  natural  sine  of  10°,  we  have 

It 
.':p  =  yr;  =  .174533  nearly.     This  value  substituted  in  the  formulcF.,  will  give 

the  sin  10°  =  .17365,  and  cos  10°  =  .98481.     The  series  converge  so  rapidly 
that  but  few  terms  are  necessary. 

1 
Ex.  5.  Develop  y  =  {a^  -\-  hx)^  hj  Maclaurin's  Formula. 

Sug's.     y  =  (a-^  -\-  bx)^,  .-.  (y)  =  a, 


dy 
dx 


.  S,  =  (a=  +  6x)    =„  +  ---_+_  _,  etc. 


*  This  notation  signifies  "  x  being  made  equal  to  Q. 


70  APPLICATIONS  OF  THE  DIFFEEENTIAL  CALCULUS. 

Ex.  6.  Develop  y  ==  \/l  -\-  a^. 

^      -,  /:; -  ^  1        ^       x^       X*       x^        bx^ 

Result,  y=Vl  +  X"'  =  (1  +  vr=)2  =l+__-+__— _+, 

etc. 


Ex.  7.  To  produce  the  logaritlimic  series. 

Solution. — This  series  is  the  development  of  y  =  log  (1  -)-  x).     Differentiating 

dv 
with  reference  to  a  system  of  logarithms  whose  modulus  is  m,  we  have,   -r  = 

dx 

m      d^y m         d^y 2m         d^y  2  -  3m  .^ 

iq:^'d^2-~(TT^'d^^'"a+^'  d^4  =  -(rqr^'^*''-     Whence(t/)  = 

^-1  =  °;  (l)  =  -  (S)  =  --  (S)  =  -'  (5^)  =  -----  -•  «^''- 

stituting  in  Maclaurin's  Formula,  we  have 

y  =  log  (1  -|-  a;)  =  7n(ic  —  gic^  +  ix^  —  ix*  +,  etc.), 
the  law  of  the  series  being  ajDparent. 

120.  Cor.  1. — Since  in  the  Napierian  system  m  =  1,  we  have 
y  =  log(l  +  x)  =  X  —  1x2  +  ^x3  —  ix^  +  ix5  —  etc. 

127*  ScH.  1. — This  formula  is  not  adapted  to  the  purpose  of  computing 
logarithms,  since  it  is  diverging  for  integral  values  of  x.  Thus,  letting 
ic  =  2,  we  have  y  =  log3  =  2  —  2  +  f  —  4  +  ^e^.  — ^  etc.,  in  which 
each  term  after  the  first  two  is  greater  than  the  preceding,  and  hence  ex- 
tending the  series  does  not  approximate  the  value  of  log  3. 

From  the  series  in  the  corollary,  however,  a  converging  series  may  be 
readily  deduced.     The  following  is  a  simple  method  : 

Substituting  — x  for  x  we  have 

log  (1  —  x)  =  — X  —  \x^  —  ^x"^  —  ^x^  —  \x^  — ,  etc. 

Subtracting  this  from  the  former,  we  obtain 

log  (1  +  ^)  —  log  {l—x)=  log  ]  \^-^  \  =2{x-^  i.^3  _},  la;--  +  W  +,  etc.) 

(1  —  X  ) 

Now  putting  X  =   ,  whence  — ^^-  =  -^~,  we  have  log = 

^  ^  2z  -i-1  1  —  x  z  z 

log  iz  +  1)  — log^  =  2^ — ^ \ 1 h . ^ \ Jt h,  etc. ), 

^^   ^    ^         ^  V22  +  1^3(22  +  l)3^5(22+l)=^7(2^+lj7^'         /' 

or  log  (2  +  1)  = 

log2  +  "li— 1 ^ \ ^ I ^ 1 — {-,  etc.  \ 

^    ^    \22  +  l  ^3(22  +  1)3^5(22  +  1)^^  7(2^  +  1)7^9(22  +  1)9^'        / 

This  series  converges  for  all  positive  values  of  z,  and  more  rapidly  as  2  in- 
creases. 

To  apply  this  formula  in  computing  a  table  of  Napierian  logarithms,  first 
let  2  =  1,  whence  log  2  =  0  + 

0/1,1,1,1,1,        1        ,        1  1        ,^\ 

V3  ^  3  •  33  ^  5  .  3'  ^  7  -3'  ^  9  •  3^  ^  11  -B-'  ^  13  .  S'-'-      15  •  3'^  ^  / 


DEVELOPMENT  OF  FUNCTIONS. 


71 


The  mimerical  operations  are  conveniently  performed  as  follows  : 


3 

2.00000000 

9 

.66666667 

9 

.07407407 

9 

.00823045 

9 

.00091449 

9 

.00010161 

9 

.00001129 

9 

.00000125 

.00000014 

l0£ 


1 

3 
5 

7 

9 
11 
13 
15 

2  = 


.66666667 
.02469136 
.00164609 
.00013064 
.00001129 
.00000103 
.00000009 
.00000001 


.69314718 


Second.     To  find  log  3,  make  z  =  2,  whence  log  3 


(-  +  - 

\5  ^  3  . 


Compuiaiion. 


^5 

5 

25 
25 
25 
25 


+ 


3-53  '  5 

2.00000000 


+ 


7-5 


log  2  + 

.  _f  -L     4-  etc.  ) 


.40000000 
.01600000 
.00064000 
.00002560 
.00000102 


.40000000 
.00533333 
.00012800 
.00000366 
■00000011 

.40546510 
.69314718 


Third. 

Fourth. 


il 


9'^3 


To  find  log  4 
find 
1 


To 

+ 


93   '    5  •  9-i   '    7-9 
Computation.       9 


log  2 

.■•.  log  3  =     1.09861228 

log4  =  21og2  =  2  X 
log  5.      Let    ^   =  4, 

2.00000000 


69314718  =  1.38629436. 
whence    log  5   =   log  4   -|- 


81 
81 
81 


.22222222 
.00274348 
.00003387 
.00000042 


log  4 


.22222222 
.00091449 
.00000677 
■00000006 

"722314354 
1.38629436 


.-.  log  5  =  1.60943790 

In  like  manner  we  may  proceed  to  compute  the  logarithms  of  the  prime 
numbers  from  the  formula,  and  obtain  those  of  the  composite  numbers,  on 
the  principle  that  the  logarithm  of  the  product  equals  the  sum  of  the  log- 
arithms of  the  factors. 

The  Napierian  logarithm  of  the  base  of  the  common  system,  10,  = 
log  5  +  log  2  =  2.30258508. 

12 S»  Cor.  2. — The  logarithms  of  the  same  number  in  different  sys- 
tems are  to  each  other  as  the  moduli  of  those  systems ;  and  the  logarithm 
of  a  number  in  any  system  equals  the  Napierian  logarithm  of  the  same 
number  multiplied  by  the  modidus  of  the  proposed  system. 


120a  ScH.  2. — To  find  the  modulus  of  the  com,mon  system  of  logaritlims. 


72  APPLICATIONS   OF  THE  DIFFERENTIAL   CALCULUS. 

we  have  com.  loff.  x  =  m  Nap.  loe:.  x,  whence  m  =  — — '-—^ — .     Now  hav- 

ing  computed  the  Napierian  logarithm  of  10,  by  the  formula  above,  and 

found    it   to    be    2.302585,    we   have   m  =   — — \     \,,   =   ,,  ^^^  ^-   = 

Nap.  log.  10  2.302585 

.43429448+. 

ISO,  ScH.  3. — To  compute  a  table  of  common  logarithms,  first  compute 
the  Napierian  logarithms  and  then  multiply  by  the  modulus  of  the  common 
system,  .43429448. 

Ex.  8.  To  ascertain  the  relation  of  the  modulus  of  a  system  of  log- 
arithms to  its  base. 

SoiiUTioN. — Developing  y  =  a^,  by  Maclaurin's  Formula,  we  have 
1  1    05-         1      a;^  1        rc^ 

!'  =  "'  =  1  +  m^  +  »-^  2-  +  m>  a—  +  m.  l^aTI  +  ^*°-     ^^'^ 
Again,  putting  a  =^1  -\-h,  and  developing  by  the  Binomial  Formula,  we  obtain 


'   .  2-3-4-5 

Expanding  and  collecting  the  coefficients  of  the  1st  power  of  x  we  find  it  to  be 

^-2-+3--4+5-6-+'^*"- 
Finally,  since  series  (1)  and  (2)  are  equal  the  coefficients  of  like  powers  of  x  are 

1                  62       53       54       55       ^G 
equal,  and  — =6  —  tt  +  t; -7--+-? ;; — hj  etc. ;  or  restoring  a  and  findmg 

m  23456 

the  value  of  m,  we  have 

1 

vn    — —  ^ 

(a  — 1)  — i(a  — l)2  +  i(a  — 1)3  — i(a  — l)4  +  i(a  — 1)^  — -L(a— l)6-f,  etc. 

131,  ScH. — To  find  e,  the  base  of  the  Napierian  system.  Since  the  log- 
arithms of  the  same  number  in  different  systems  are  to  each  other  as  the 
moduli  of  those  systems,  we  have 

com.  loge  :  Nap.  log  e(=  1)   ::  .43429448  :  1. 
.•.  com.  loge=  .43429448,  and  e  from  the  table  of   common  logarithms, 
which  we  have  shown  how  to  compute,  is  2.718281+. 

Ex.  9.  To  develop  y  =  a"",  i.  e.  to  produce  the  exponential  series. 

X  X'  x^ 

Result,    1/  =  a-  =  1  +  log  a  J    +    log^  aj-^    +   log^  a^  ^  ^  + 

x^ 

X 

132.  ScH. — ^If  a  =  e,  the  Napierian  base,  this  becomes  q/  =  e'  —  l-{--  + 

x-^     ,        a;3        ,  X*  ,  x^  ,       . 

\~,  etc. 

12^1..2-3^1-2-3.4^1-2.3.4-5^ 

If  a:  =  1,  we  have 


DEVELOPMENT  OF  FUNCTIONS.-  73 

y==^_,_2  +  -  +  --  +  ^-g--  +  2.3.4.5  +'  ^t«- '  a  ^^^^^  for 
finding  the  Napierian  base,  although  the  series  converges  slowly. 


Ex.  10.  Develop  y  =  tan  ^x. 

SoiiUnoN. — Differentiating  --  z=  - — ; ,  which  by  division  becomes  ~  =z\  — 

dx       1  +  a;2  •'  dx 

a;2  +  jc*  —  icG  -f  a;«  —  x'"  +,  etc. 

Differentiating  successively,  -r-^  =  —  2a;  +  4a;''  —  Gx^  +  Sx^  —  lOx^  4-,  etc. 

dx'-^ 

-!|  =  —  2  +  3.4a;2  —  5.6a;^  +  7-8a;6  — 9-10x8+,  etc. 

-^  =  2.3-4rc  —  4-5-6x3 4- 6-7.8x5  —  B-Q-lOx^-f ,  etc. 

^  =  2-3.4  — 3-4.5.6x2  +  5.6.7-8x''—7-8.9-10xe-h,  etc. 

^.  =  —  2-3.4.5-6X+4  5-6.7-8x3  —  6.7-8-9.10x5 -fete, 
ax"  ' 

menee  („  =  tan-.<,  =  0,  (|)  =  X,  (g)  =  0,  (^^)  =  -  2,  (g)  .  0, 

[y4)  =  2-3-4,  f-— jzizO,   etc.     Introducing  these  values  into  Maclaurin's 
Formula,  we  have  2/  =  tan— 'a;  =.  x  —  \x'^  -\-  \x^  —  ^x"^  -f"  9^^  —  A^"  +>  etc. 


ScH. — By  means  of  this  development  we  are  enabled  to  find  the  value 

It  Tt 

of  It.     Thus  let   y  =  45°  =  j,  whence    x  =  1,  and    we  have  y  =  —  = 

*       11        1        1|1       1,1         1,1  1,1         1,. 

tan-l=l--  +  -_-  + ---  +  ----+--_+,  etc. 


133,  Pvop, — Though  Maclaurin's  Formula  is  applicable  to  a  very 
great  variety  of  forms  of  functions  of  a  single  variable^  it  will  not  develop 
ALL  such  functions. 

The  truth  of  this  theorem  will  be  substantiated  if  we  can  present  examples  of 
functions  of  a  single  variable  which  the  formula  will  not  develop  properly.  This 
we  proceed  to  do. 

1  . 
Ex.  1.   Show  that  y  =  x^  i^  not  properly  developed  by  Maclaurin*s 

Formula. 


Solution. — From  y  =  x  ,  we  have  -r-  =  — ;,  -rp^  = -,  etc.     Hence  (v)  =  0, 

ax      „   5  ax2  I  \s/        J 


f  -^  j  =  -  =  00,  (  —  j  =  —  oc,  etc .    Substituting  these  values  in  the  formula 
have  y  ==  X    =  0  -4-  cox  ^ —  co'----f-,  etc.     Such  results  as  these  will  be  simply 


we 


74  APPLICATIONS  OP   THE  DIPPERENTIAL   CALCULUS. 

unintelligible  to  the  learner  at  first.     But  in  this  case  it  is  easy  to  see  that  the  de- 

yeloijment  will  consist  of  pairs  of  terms  of  the  same  general  form  as  (x>x  —  oo  — , 

2t 

To  ascertain  just  what  is  to  be  understood  by  this  binomial,  let  us  restore  the 

values  of  oo  as  they  were  before  x  was  made  equal  to  0,  only  using  x   to  indicate 

the  X  that  is  0.     We  then  have '- '—,  or ^-1-.     Now  as  x  =  0, 

2ic'^  4.x' '^  8a;' ^ 

this  becomes  —  ^ ,  which  is  oo  for  all  values  of  x  except  0,  and  indeterminate  for 

that.     In  like  manner,  it  may  be  shown  that  each  succeeding  pair  of  terms  equals 

00  .  Hence  we  have  the  absurd  result  that  ?/  =  x^  =  oo,  for  aU  values  of  x,  since 
the  development  should  be  true  for  all  values  of  the  variable. 

Ex.  2.  Show  that  y  =  log  x  is  not  properly  developed  by  Maclau- 

rin's  Formula. 

Sug's. — The  result  is  similar  to  the  preceding  except  that  the  first  term  is  oo  in 
this  case.  Each  succeeding  binomial  may  be  seen  to  be  oo,  as  in  the  former  case. 
Hence  we  have  the  absurd  result  that  y  =  log  a;  =  oo  for  all  values  of  x. 

1 
Ex.  3.  Show  that  y  =  cot  x,  and  y=^  a"  are  not  properly  developed 

bv  Maclaurin's  Formula. 

134,  ScH. — The  occasion  of  the  inapplicability  of  Maclaurin's  Formula, 
in  such  cases  as  just  given,  is  the  fact  that  the  form  of  the  function  is  such 

that  the  coefficients  -f-,  — ^,  etc.,  or  the  function  itself,  or  both,  become 
a.v   ax' 

infinite  for  x  =  0,  which  is  contrary  to  the  hypothesis  upon  which  the 
formula  was  produced.  Whether,  in  such  cases,  the  failure  to  develop  cor- 
rectly by  this  formula  is  due  to  the  fact  that  the  particular  function  is  in- 
capable of  any  development,  or  whether  it  is  simply  because  it  will  not 
develop  in  the  particular  form  assumed  in  this  theorem,  does  not  as  yet  ap- 
pear, and  our  limits  forbid  our  entering  upon  the  question. 


TATLO?.'S   FORMULA, 

135.  Def. — Taylor's  Formula  is  a  formula  for  developing  a 
function  of  the  sum  of  two  variables  in  terms  of  the  ascending 
powers  of  one  of  the  variables,  and  finite  coefficients  which  depend 
upon  the  other  variable,  the  form  of  the  function,  and  its  constants. 

136.  Lemina* — Jff^  u  =  f  (x  +  y)  the  partial  differential  coefficients 

du       _  du 

--  ana  -z—  are  equal. 
dx  dy  ^ 

Dem.— Having  u  =  f(x  -f-  y),  if  x  take  an  increment,  we  have  u  -f  dxu  = 
f(x  -^  dx  -{-  ij)  =f[{x  -\-  y)  -\-  dx]  ;  whence  d^u  =  f  [{x -{- y)  -{- dx]  —,f{x  -f-  y). 
Again,  if  y  take  an  increment,  we  have  u  -\-dyU  =f{x-\-y-{-dy)  =fl{x-'ry)-^dy]; 


DEVELOPMENT  OF  FUNCTIONS.  75 

whence  dyu  =f{{x  -}-  y)  -\-  dy}  —fix  +  y)'  Now  the  form  of  the  values  of  dxU. 
and  dyU,  as  regards  the  way  in  which  x  and  y  are  involved,  is  the  same  ;  hence, 
if  it  were  not  for  dx  and  dy,  they  would  be  absolutely  equal.  Passing  to  the  differ- 
ential coefficients  by   dividing  the  first  by  dx  and  the  second  by  dy,  we  have 

du      /[(.r  +  y)  -\-dx-]—f[x-\-y)  du  _/[(x  +  ?y)  +  dy]  — .Ax-f  y) 

= — ,   ana    -J-  — .     x)Ut, 

dx  dx  dy  dy 

in  differentiating,  the  differential  of  the  variable  enters  into  every  term  ;  hence 
f[{x  +  2/)  +  dx]  — fix  -j-  y),  as  it  would  appear  in  application,  would  have  a  dx  in 
each  term  which  would  be  cancelled  by  the  dx  in  the  denominator  in  the  coef- 
ficient, and  —  would  be  independent  of  dx.  In  like  manner  —  is  independent  of  dy. 
dx  (ly 

Hence,  finally,  as  these  values  of  the  partial  differential  coefficients  are  simply  func- 
tions of  {x-\-y),  of  the  same  form,  and  not  involving  dx  or  dy,  they  are  equal,    q.  e.  d. 

ScH. — The  substance  of  this  demonstration  is  that  the  values  of  the  dif- 
ferential coefficients  depend  upon  the  form  of  the  function,  and  are  inde- 
pendent of  the  increment  of  the  variable.  Therefore  Avhen  the  form  of  the 
function  is  such  as  to  give  to  the  partial  differential  coefficients,  the  same 
form  with  respect  to  the  variables,  the  coefficients  are  equal.     But  suppose 

.       du        f[{x-\-dx)if]—f{xy)        f[xy  +  ydx)  —  f{xy)  ^ 

we  have  u  =.  fixy).     -—  =  -^ ; =  -^— ; 

dx  dx  _  dx 

and  ^  =^^''^^  +  "^^^  -■^^''^^  ^^^^'"  +  ""'^-'^  -A^y),     In  these  coef- 
dy  dy  dy 

ficients  we  see  that  the  form  is  not  such  as  to  involve  x  and  y  in  the  same 

way  ;  hence  they  are  not  necessarily  equal.     A  few  examples  will  render 

the  truth  of  the  lemma  more  clear, 

Ex.  1.   Given  u-=  (^  +  y)"*  to  show  that  the  partial  differential 

coefficients  are  equal. 

-,.      -,      du  ,  V     ,         -,  du  ,      .      N„  , 

Besults,  —  =  m{x  +  y)      ,  and  —  =  m{x  +  y)      • 

Ex.  2.  Given  u  =  log  {x  -{-  y)  to  shovr  that  the  partial  differential 

„  du  1  du  1 

coefficients  are  equal.  Results,  -r-  =  — ■ — ,  and  -7-  =  — ■ — :. 

•^  dx       X  -\-  y  dy       X  -\-  y 

Ex.  3.  Given  u  ==  tan~^(a;  -\-y)  to  show  that  the  partial  differential 

coefficients  are  equaL 

^    ^      du                  1  ^  du  1 

Besults,  --  = r-,  and  3- 


dx       1  +  (^  +  yY  dy        1  +  {^'  +  yY 

-j  ,  show  that  the  partial  differential  coefficients 

are  not  equal. 

^      ,      du        mx'^~^        ^  du  7??.r'"v"'~^  „  _™  , 

Results,  ~  =  — -— ,  and  -- = "^ —  =  —  rnx'^y-^'K 

'  dx  2/  dy  ?/2">  ^ 

Ex.  5.  Given  u  =  log  (xy)j  show  that  the  partial  differential  coef- 
ficients are  not  equal  Results^  -—  =  -  and  --  =  -. 
*  ax       X  dy        y 


76  APPLICATIONS  OF  THE  DIFFEKENTIAL  CALCULUS. 

137*  I*VOh, — To  produce  Taylor's  Formula. 

Solution. — Let  u  =f{x  -\-  y)  be  the  function  to  be  developed.  It  is  proposed 
to  discover  the  law  of  the  development  when  the  function  can  be  developed  in  the 
form 

u  =f{x  ^y)  =  A  +  By+0!^  +  Dy^^Mj*  +,  etc.,     (1), 
in  which  A,  B,  C,  etc.,  are  independent  of  y,  and  dependent  upon  x,  the  form  of 
the  function,  and  its  constants. 

Differentiating  with  respect  to  y,  remembering  that  as  A  is  independent  of  y  it 
will  disappear,  and  that  as  the  factors  B,  C,  B,  etc.,  are  likewise  independent  of  y, 
they  are  to  be  regarded  constant,  we  have 

^  =  5  +  2Q/  4-  3i)2/2  +  4Z2/3  +,  etc.     (2). 

Again,  differentiating  with  respect  to  jc,  we  have 

du        dA*    ,    dB  dC'      ,    dB  ^    ^ 

-r  =  T-    -\-  -T-y  +  T-y-  +  T-y^  +»  etc.     (3). 

dx        dx     ^  dx^  ^  dx^    ^  dx^    ~'  ^  ' 

Hence  by  (136) 

S  +  2C,  +  ZDr-  +  W  +,  etc.  =  ^  +  ^2,  +  ^-3  +  ^^y,  +,  etc. 

Now,  by  the  theory  of  development  by  indeterminate  coefficients,  the  coefficients 
of  like  powers  of  y  are  equal,  and  we  have 

B  =  ^A,    2a  =  ^,    3i>  =  5^?.    4£'=^,    etc. 
dx  dx  dx  dx 

But  as  (1)  is  true  for  all  values  of  y,  we  may  make  y  =0,  whence  A  =f(x)  =  u, 

du 
letting  u  represent  the  value  of  the  function  u  when  y  =  0.     Hence  B  ==-—, 

(X*C 

r  —-i-  —I  ^^■^'  ^  _<?!^1   7)_1^_1   1  .\  dxV  ^  ^'  J_       ,  . 
2  d£^  ""  2      dx      ~  dx-^  2'  3dx~  2'S      dx       ~  dx^  2  •  3'  ^^    "^ 

T.      d^u        1 
like  manner  £J  =  — — -  -— - — -,  etc. 
dx*    2  •  3  •  4 

Substituting  these  values  of  A,  B,  C,  B,  etc.  in  (1),  we  have 

,     ,  ,    ,   du'  y   ,   d^u     y^     ,   d^u        y^      .   d^u         y* 

u=Ax  +  y)  =  u  +_--4.__^+_-----^+  — 3-^-3-^+,etc., 

which  is  the  formula  sought.  • 

138,  ScH. — Taylor's  Formula  develops  uy  =  f{x  -j-  y)  into  a  series  iii 
which  the  first  t&)^m  is  the  value  of  the  function  when  y  =  0  :  the  second  term 
is  the  first  differential  coefficient  of  the  function  when  y  =  0,  into  y  ;  the 
third  term  is  the  second  differential  coefficient  of  the  function  when  y  =  0, 

into  -^ —  ;  etc.,  etc. 
1-2'         ' 

Ex.  1.  Develop  u=  {x  -\-  y)"^  by  Taylor's  Fornmla,  and  thus  de- 
duce the  Binomial  Formula. 

Solution. — Making  ?/ =  0  we  have  u'  =  x^.     Differentiating  u  =  x'",  succes- 

sively,  we  ootam  — -  =  ?n.r"'— \  - —  =  mim — l)x" — ,  - —  =  tmm, —  l)(m  —  2)x'"~^ 
dx  dx'^  dx:^ 

*  TbiB  is  the  proper  form,  since  A,  B,  C,  etc.,  are  functions  of  x. 


DEVELOPMENT   OF  FUNCTIONS.  77 

d'*u' 

— -  =  m{m  —  l)(m  —  2)(m —  3)a;'"— S  etc.     Substituting  these  results  in  Taylor's 

Formula,  we  have 

,     ,  ,  ,      ,   m(m  —  1)        „       ,    m(m  —  l)(m  —  2) 

XL  z=  (X  -f-  2/)'"  =  ^"'  +  vnx'^-^y  -j ^ — -x'^-^-y"-  -j -—^ -a^-^i/s  4- 

u  2  •  o 

■m{m  —  l)(m  —  2)(7n  —  3)        .     .  ,  ,.,.,,..         .  ,  .^ 
-— - — a;"'— *V^  4->  etc.,  which  is  the  Binomial  Formula. 

Ex.  2.  Develop  u  =  log  {^x  -\-  y). 

Sug's. — This  being  a  function  of  the  sum  of  two  variables,  we  apply  Taylor's 

T,         ,         ,        ,  du        1  dfiu  1    d?u'       2 

Formula,    w    =  logx,  — —  =-,  - —  = , =  ~,  etc. 

ax,       X    ax?-  's?'  dx,^        'x? 

Henceu  =  log(.r  +  2/)=log^  +  |-^|4-£-£+,etc. 
Ex.  3.  Develop  u  =  a='+^ 

Besult,  u  =  a\l  +  log  a  y  +  -^y^2/^  +  -|^2/'  +>  etc.). 
Ex.  4.   Develop  w  =  sin  (^  +  2/)- 

„     ,         ,  .         du  d^u'  .        d^u 

SuG  s.     u    =  sm  X,  — —  =  cos  a;,   - —  7=  —  sm  x,   - —  ==  —  cos  x,  etc.     Hence 
dx  dx'^  dx^ 

V  .  y^  /j/3  y4 

u  =  sm  {X  -\-  y)  =  sm  x  -j-  cos  a;^  —  sm  x- — -  —  cos  x    '^     -  -\-  sin  .r- — '  -f- 

,in.,(l    -    ^-^    +    1-2^-4    -    1.2./4.5.6    +'    "*"•>    + 

"'''^^(2'  -  T^  +i.2.3'.4."5  -  l.2.3.4'.5-6-7  +'  '''''•'  =  ^'"*'=<>«2'  +  <=os.rsiny, 
since  the  series  in  the  parentheses  are  equal  respectively  to  cosy  and  sin 2^ 
{124,  Ex's  3,  and  4). 

Ex.  5.  Develop  u  =  cos  {x  -\-  y). 

yi  yi  yB 

^esult,u=cos{x-^y)=Gosx{l—~-\-^^^^  —  ^^^^^^^-\-,etG.) 

-  ^^<y  -  rf:  3  +  T^iir^  - 1.2.3.4.5.6.7  +'  "*"-^  == 

cos  X  COS  2/  —  sin  a?  sin  2/. 

Ex.  6.   Develop  w  =  sin  (x  —  y),  and  also  u  =  cos  (a;  —  y). 
Results,    u  =  sin  {x  —  y)  =  sin  x  cos  ?/  —  cos  x  sin  2/,   and  u  = 
cos  (^  —  2/)  ==  cos  ar  cos  2/  +  sin  x  sin  2/. 

Ex.  7.    Develop   u  =  (a;  +  2/)^    also   w  =  (a?  —  i/)^,  by  Taylor's 
Formula. 

Ex.  8.  Develop  w  =  (x  —  j/)"*. 


78  APPLICATIONS  OF  THE  DIPFEBENTIAL  CALCULUS. 

130,  Taylor's  Formula  is  mucli  used  for  developing  a  function  of 
a  single  variable  after  the  variable  has  taken  an  increment.  When  so 
used  the  increment  may  be  conceived  as  finite  or  infinitesimal,  only 
so  that  it  be  regarded  as  a  variable. 

Ex.  1.  Given  y  =  log  x,  to  find  y'  which  represents  the  value  of  y 
after  x  has  taken  the  increment  h. 

Solution,     y'  =  log  {x  +  '*')>  whicli  developed  by  Taylor's  Formula  gives 

S  ""  2x2  "^  3^  "~  i^  "^'  ^*^' }  "^  ^^^^^  *^®  modu- 
lus of  the  system  of  logaritlims. 

ScH.  — If  h  be  considered  infinitesimal  with  respect  to  x,  so  that  we  have 
Ji  =  dx,  we  may  drop  all  the  terms  within  the  parenthesis  except  the  first, 

and  write  y'  =  log  re  -j '-.     This  is  the  consecutive  state  of  the  function 

y  =  log  X.     Hence  subtracting  the  latter  from  the  former  we  have  y'  —  y  = 

dy  =  d  log  X  =  .     This  result  is  as  it  should  be,  in  accordance  with  the 

X 

rule  for  differentiating  a  logarithm. 

Ex.  2.  Given  y  =  3x  —  2x^  —  5,  to  find  y',  which  represents  the 
value  of  the  function  after  x  has  taken  the  increment  h. 

Result,  y  =z  3;r  —  2^3  _  5  4_  (3  _  Qoc^)h  —  llx^  — 12^  =  3a;  — 

2^3  _  5  _|-  (3  _  6a72)/i  —  6a7/i2  —  2h\ 

ScH. — This  result  may  be  easily  verified  by  direct  substitution.     Thus, 

y'  =  3{,r  +  h)  —  2(.r  4-  hy  —  5.     Expanding,  y'  =  Sx  +  3h  —  2x^  —  Gx^h  — 
Qxhi  _  2^3  _  5  =,  3^  _  2a73  —  5  +  (3  —  6^2)  7j  —  6x71^  -  2h\ 


140,  Prop. — Though  Taylor's  Formula  gives  the  general  form  of 
the  development  of  a  function  of  the  sum  of  two  variables,  there  are 
sometime.^  particular  values  of  one  or  the  other  of  the  variables  for  which 
the  development  is  not  true. 

We  will  illustrate  this  proposition  with  a  few  examples. 

JL 

Ex.  1.  Develop  u-=i  {x  -\-  y  —  a)^  by  Taylor's  Formula,  and  show 
that  the  development  is  false  when  x  =^  a. 

,       ,  d    du         ,^  -h  1  d^u  -f 

Solution,     u  =  (a;  —  a)  ,  ——==  i{x — a)      = j,  ——^=  —  ^{x  —  a) 

dx  ^  a  dx^ 

2{x  —  a) 

: ,  etc.     Hence  substituting  in  Taylor's  Formula, 


f  dx^        „,„       ^J 


4(0:  — a)^  ^{x—ay 


DEVELOPMENT  OF  FUNCTIONS.  79 

i  i  y  y^  ^v^ 

we  have  w  =  (a;4-2/  —  «)    =  (* — <*)    -\ 1 ^H ^ — .etc. 

2{x  —  af      8(x  —  a)''^      m.x  —  af 
Now,  no  absurdity  appears  in  this  series  for  general  values  of  a; ,  but  for  x  =  a 

the  series  becomes  oc,  while  (x  -^  y  —  a)  =  2/  ?  ^or  the  same  value.  But  by  hy- 
pothesis X  and  y  are  independent  and  the  development  should  be  true  for  any 
value  of  y  irrespective  of  the  value  assigned  to  x.     Hence  the  conclusion  that  for 

X  =  a,  y^  =  oc  is  contradictory  to  the  hypothesis,  and  false. 

ScH. — It  is  evident  that  any  form  of  function  which,  when  developed  by 
this  formula,  gives  a  factor  of  the  form  [x  =F  «)"*  in  the  denominator  of  any 
term  in  the  development,  will  afford  an  instance  similar  to  the  above,  and  the 
development  will  not  be  true  for  x  =  ±  a,  since  for  this  value  [x  =+:«)"'=:=  0, 
and  the  terms  in  the  denominators  of  which  it  occurs  will  reduce  to  oo. 

7. 

Ex.  2.  For  what  value  of  x  is  the  development  of  w=  (^x-\-y-{-b)''^ 
by  Taylor's  Formula,  untrue  ?  Ans.,  x  =  —  h. 

Ex.  3.  Required  the  value  of  the  function  after  x  has  taken  an  in- 

3. 

crement  h,  when  y  =  h  -\-  (a;  +  c)^  +  (-^ —  ^)^'  For  what  value  of  x 
does  the  development  fail  ? 

Result,  y'  =  h  -^  (x  ^  cy  ■}-  {x  —  a)^  +  [2(^  +  c)  +  ^{x  —  aY]h  -f 

[2  +  |(^  _  «)-i]|'  _  !(:.  -_  a)-t^  +,  etc. 

y'=  cc  when  x  =  a,  and  hence  the  development  fails  for  this  value. 

ScH.  1. — If  h  =  dx  the  above  development  is  true  for  all  values  of  x,  for 

3  1 

then  we  have  y'  =  b -\-  {x -\-  c)2  -f  (.r  —  a^  -{-  [2{x  +  c)  +  f  (.t —  a)  ]h,  which 
is  the  same  as  would  be  obtained  by  substituting  x  -j-  h  for  x  in  the  first 
state  of  the  function  and  developing,  and  then  making  h  =  dx,  and  dropping 
the  higher  powers  of  h.  For  x  ^  a  this  becomes  y'=  Z>  +  (a  -f  c)^  -f- 
2 (a  -f  c)h,  which  is  as  it  should  be,  since  for  x  -{-  h  =  a  -{-  h,  y'  =  h  -\r 

{a  -\-  h-]-  cY  +  [a  -\-  h  —  af  =  5  +  «2  +  2a^  +  2ac  +  7^2  +  Ihc  +  c^  +  ]{'  = 
(dropping  higher  powers  of  h)  b  -\-  a^  -\-  lah  +  2«c  +  2hc  -f-  c^  =  5  -j- 
(a2  +  2ac  +  c2)  +  {2ah  +  2c^)  =  6  +  (a  +  c)2  +  2(a  +  c)h. 

ScH.  2. — ^It  will  be  observed  that  when  Maclaurin's  Formula  fails  to  give 
the  true  development  of  a  function  it  fails  for-  all  values  of  the  variable  ; 
but  when  Taylor's  fails  it  is  only  for  particular  values,  the  general  develop- 
ment being  still  true. 

GENEKAii  Scholium. — There  are  many  other  important /brmM^cc  for  the 
development  of  functions,  but  the  prescribed  limits  of  this  volume  pre- 
clude their  presentation. 


80  APPLICATIONS   OF   THE  DIFFERENTIAL   CALCULUS. 

SECTION  11. 

Evaluation  of  Indeterminate  Expressions, 

14:X,  The  following  forms  are  called  The  Indeterminate  Forms,  viz., 

0    00 

-,  —   0  X   00,   00  —  00,  0°,   oo«,  1". 

U      00 

Whenever  an  expression  assumes  any  one  of  these  forms,  the  impor- 
tant question  to  be  determined  is  whether  it  is  ideally  indeterminate, 
for  it  often  happens  that  the  indetermination  is  only  apparent. 

Of  these  forms,  -  is  the  fundamental  one,  to  which  all  the  others 

can  be  reduced. 

IiiL. — That  -  is  an  indeterminate  form,  is  readily  seen  when  we  observe  that 

the  divisor,  0,  multipUed  by  any  finite  number,  produces  the  dividend,  0. 

We  may  show  that  each  of  the  other  forms  can  be  reduced  to  the  first,  and  hence 

that  they  are  indeterminate  forms.     Thus,  let  a  represent  a  finite  quantity  ;  then 

a  a 

0         00^5        a        0        O^^^oo.         .^^       -xr  iu 

=  — .  But  -  =  -  X  -  =  zz-  That  —  is  an  indeterminate  form  may  also  be 
a         CO  a        0        a        0  ao 

0  0 

seen  directly ;  since  one  infinity  may  be  any  number  of  times  another,  and  the 

symbols  oo  do  not  mean  that  numerator  and  denominator  are  the  same  infinity. 

Again  OX   '^  =  -  X  t:  =  7{,   «  being  any  finite  quantity.     Also  oo  —  oo  is  inde- 
a      0       0 

terminate,  since  the  difierence  between  two  infinities  may  be  any  quantity  what- 
ever.    Taking  0"  and  passing  to  logarithms,  we  have  0  log  0  =  0( —  oc)  =  —  0  X  oo, 

which  has  been  shown  equal  to  -.     Finally,  applying  logarithms  to  ooo,  and  1 ",  the 

former  becomes  Ologoo  z=  0  X  cc,  and  the  latter  oologl  =  oo  x  0.       ^ 

142,  The  apparent  indetermination  often  occurs  from  the  intro- 
duction of  some  hypothesis  which  introduces  a  factor  0,  into  both 
terms  of  the  fraction. 

^3  jp3 

III. — What  is  the  value  of when  a;  =  a  ?    Making  x  =  a  reduces  the 

a  —  X 

0  (j3  X^         . 

expression  to  -  ;  whence  it  would  appear  that  is  indeterminate  for  x  =  a. 

u  a  —  X 

^3 a;3 

But  that  such  is  not  the  case  is  evident,  since =  a^  -{-  ax  -4-  x'^  which  =  3o2 

a  —  X 

when  x  =  a.  This  apparent  indetermination  arises  from  the  fact  that  the  hypoth- 
esis x  =  a  introduces  a  factor  0  into  numerator  and  denominator.  This  factor 
being  divided  out,  the  true  value  is  seen.    But  it  is  not  always  easy  to  discover 


i 


EVALUATION   OF   INDETEKMINATE   EXPRESSIONS.  81 

the  factor  which  becomes  0,  so  as  to  be  able  to  cancel  it ;  hence  the  necessity  of 
some  general  method  of  procedure. 


f(x) 
14:3.  I^vob, — To  evaluate  y  =      ,  \  for  x  ==  a,  when  for  this  value 

of  the  variable  the  /unction  assumes  the  form  —. 

Solution. — Let  y'  be  the  function  when  x  has  taken  an  increment  h,  so  that 

f(x  -\-  h) 

y'  =  — ■ — —,     Developing /(aj  -f-  /*,)  and  wix  4-  h)  by  Taylor's  Formula,  and- for 

cp{x  -\-  h) 

simplicity  using  f  {x),  f'{x), (p\x),  (p"{<^),  etc.,  for  the  coefficients,  we  have 

fix)  +  f{x)^  +  /"(^)r^'-77  +,  etc. 
I  ex  -^-  ft  I 

y 


fix  +  /n    _' '•^^  -r  J  v-^^i  -r  J    v-^^i  .  2  ^ 


^'    ^  <p{x)  +  cp{x)--{-<p   {x)—--{-,  etc. 

But  by  hypothesis,  when  x  =  a,  fix)  and  (p{x)  each  equals  0.     Hence  dropping 
these  terms  and  dividing  by  h,  we  have 

fix)   +  /"(^)i^-2  +'  ^*^- 

y'      _     _ 

Now  making  h  ==' 0,  whence  w'  becomes  y,  there  results  y  =  ^—r  =  '   ,,   ,,  as  the 

"^       <p{x)       <p{a) 

value  of  the  function  for  x  =  a. 

If,  however,  ^ —  =  -,  we  can  drop  the  first  two  terms  of  A,  and  dividing  by 
ip' {a)       0 

h\  making  /t  =  0,  and  x  =  a,  we  have  y  =  — -- — , 

^  ^       (p  {a) 

Thus  we  can  continue  to  replace  f{x),  and  <p[x)  by  their  successive  differential 
coefficients  until  a  pair  is  reached  which  do  not  hotk  reduce  to  0  for  x  =  a.  The 
last  result  will  be  the  true  value  of  the  symbol. 

Ex.  1.  Given  y  = '-,  to  evaluate  the  expression  for  x  =  0. 

SuG.  f{x)  =  since,  and  ^(x)  =  x.  f\x)  =  cos-r,  and  ^'(a;)  =  1.  .-.  y  = 
cosajaj^o*       1       - 

Ex.  2.  Given  y  =  — ^~,  to  evaluate  for  x  =  1.  ^ 

^        X  —  1 


Tor  ^  =  1,  2/  ==  1. 


j^5 ]j_ 

Ex.  3.  Given  y  == — ,  to  evaluate  for  ^  =  1. 

X — 1 


Eor  X  =  1,  y  =  5. 


*  Tliis  subscript  signifies  •' »  being  =■  to  0." 


82  APPLICATIONS  OF  THE  DIFFEKENTIAL  CALCULUS. 

Ex.  4.  Show  that  if  ^  ==  0,  v  =  =  lopf  r- 

^  X  °  b 

^nx gjia 

Ex.  5.  Show  that  if  x=  a,  y  =  -. r-  ==  oo. 

SuG.    f{x)  =  n&^,  and  q}'{x)  =  six — a)'—\     Hence /'(a)  =  ne"",  and  <p'(a)  =0. 

Ex.  6.  Show  that  if  j;  =  o,  v  = =  t:- 

SuG. — The  first  and  second  differential  coefficients  of  both  numerator  and  de- 
nominator reduce  to  0  ;  but  f"'{x)  =  cosjc,  and  €p"'{x)  =  6.    Hence  for  a;  =  0, 

cos  X 1 

^  ~  ~6~  ^  6" 

g^ p — ^ 2^ 

Ex.  7.   Evaluate  y  = -. for  a;  =  0.  yx=  o  ==  2. 

Ex.  8.  Evaluate  y  = — -  for  x  =  l.  y^^i  =  0. 

{1  —  x)^ 

1  r*os  ^ 

Ex.  9.  Evaluate  y  = for  x  =  0.      '  y^.,  =  h 

3 

Ex.  10.  Evaluate  y  = — —  for  a;  =  a. 

{a  —  x)'^ 

SuG. — For  X  —  a,  the  first  differential  coeflBcients  of  both  numerator  and  denom- 
inator reduce  to  0,  and  all  succeeding  ones  reduce  to  oo .  Hence  we  see  that  for 
x  =  a  the  development  of  these  functions  by  Taylor's  Formula  is  not  true.   More- 

0        00 
over,  if  it  were  true,  we  should  but  exchange  the  symbol  ~  for  — .     In  this  case, 

however,  it  is  easy  to  see  the  factor  which  gives  the  expression  the  indeterminate 

3  3  3 

form.    It  is  (a  —  x)^.     Cancelling  it,  y  =  {a  +  a;)J=a  =  (2a)^. 

Ex.  11.  Evaluate  y  = .  for  x  =  a. 

V  x'^  —  a2 


Sttg. — This  example  is  like  the  preceding.     But  dividing  by  \/x  —  a,  we  have 

Nv/^_|_yci  Nj2\/a  1 


\^x  —  a  ^  \/^  -\-  \/c 


EVALUATION   OF  INDETEEMINATE   EXPRESSIONS.  83 


of  the  variable  the  function  assumes  the  form  — , 


144,  JProb, — To  evaluate  y  =  -^"—for  x  =  a,  when  for  this  value 


Solution,    y  =  — -r  =    -^  =  ^,  when/(a;)  and  (p{x)  are  each  oo.     Now  apply- 


(p{x) 


ing  to  y  =      '  ■■  the  method  of  the  preceding  problem,  we  have 
(p'^x)        J_        _    f{x)  f{x)    ^  l(p{x)-\2' 

Ax)  Lt\m' 

ffx) 

Dividing  the  first  and  last  members  by  — -— ,  we  have 

cp'{x)        fix)         ,  /(«)         fix) 

1  =  -^V  X  ^^-T ;   whence  y  =  '^—  =  ^~. 
f  {x)         <pix)  "         cp{x)        <p  [x) 

Therefore  the  process  in  this  case  is  the  same  as  in  the  preceding. 
Ex.  1.  Evaluate  y  =  -^—  for  x=  oo, 

1 

<-■  a;  1  1  ^ 

SuG.     2/^=„  =  ■  =  = =  0. 

rw;"— 1        nx^        n  oo" 

Ex.  2.  Evaluate  y  =  ^f-^  for  x  =  0. 

cot  X 

1  1 

^''^-    y-'  =  -cosec^g;  = r  =  -  -^  =  0-    Therefore  differentiating 

sin- a; 

2  sin  X  cos  ic       0       ^ 
agam,  y;,=o  = ^ =  -  =  0. 

j_ lofil"  X 

Ex.  3.  Evaluate  y  == 5_  for  ^7=  oo,  V,     =0 


X 

Ex.  4  Evaluate  y  =  —  for  a?  =  oo. 

SxJG.— As  the  successive  differential  coefficients  continue  to  be  oo  for  a;  =  oo  until 
we  reach  the  nth,  we  differentiate  n  times  and  obtain     ^^(^  — 1)(^  — 2)  -  -  »  3-2.1 

n(n  —  l)(n  —  2)  -  -  -  3  •  2  •  1 
=         " \ =  0,  when  x  =s  oo. 


84  APPLICATIONS  OF  THE  DIFFERENTIAL  CALCULUS. 

Ex.  5.  Evaluate  y  == for  ^  =  0.  y^^  =  -— . 

cot- 

T-,     «    -r^     ,     ,  lo2ftan(2a7)  , 

Ex.  6.  Evaluate  y  =  -^ ^ — -  for  ^  =  0.  t/x_o  =  1. 

log  tan  X 


14:S»  JPvoh, — To  evaluate  j  =f(x)  x  (p[x)for  x=  a,  when  for  this 
value  of  the  variable  the  function  assumes  the  form  0  x  oo. 

Solution. — Since  the  reciprocal  of  that  function  of  x  which  becomes  oo,  is  0, 

fix)        0 
we  may  write  y  =f{x)  X  <p(^)  =  ^--= —  =  t:,   for  x  =  a,  f{x)  being  0,  and   cp[x) 

(p(x) 

fix) 
being  oo.     Therefore,  putting  the  expression  in  the  form  y  =  ^ — ,  it  may  be 

-  (p{x) 
treated  as  the  first  case  (^14:3). 

Ex.  1.  Evaluate  y=^^  sin  —  for  a;  =  oo. 

A 

.     a 
sm  — 

SuG.     Since  for  ic  =  oo,  2*  =  oo,  and  sin  —  =  0,  we  write  y  =  2^  sin  — =  — -     . 

'2^  ^  2^        2—^ 

Whence  replacing  the  numerator  and  denominator  by  their  difierential  coefficients. 

—  a2— ^  log  2  cos  — 

tre  nave  y  = — \ =  a  cos  —  :=  a,  when  a;  =  oo. 

^  ~  2-^  log  2  2^ 

7tX 

Ex.  2.  Evaluate  y={l  —  x)  tan  -—  for  jj  =  1. 

SuG. — Since  tan  -—  =  oo  when  x  =  l,   we  write  y  =         ■'    =  — — ^  (diflferen- 

A  J.  TtOO 

cot  — 

TtX  2 

tan  — 

—  1  2  2 

tiating)  =  — ^— — — —  =:  ■  r=  — ,  when  a  =  1. 

—  —  cosec2  — -        7t  cosec2  — 

A  a  it 

1 

Ex.  3.  Evaluate  y=^e  sin  x  for  ;r  =  0. 

1  .  IX 

„                    z  .            sin  a;        cos  cc  „  -  ,  7       «  ,  « 

SxJG.     y  =  e^  sm  jc  = =  .  =  x-e  cos  x  =  ic^  e""  =  0  X  oc,  when  a;  =  0. 

1 

Were  we  to  repeat  the  process  upon  x-e^  we  should  find  that  its  form  would  re- 


EVALUATION   OF   INDETEKMINATE   EXPKESSIONS.  &5 

1  ~         6* 

main  the  same.     But  put  -  =  z,  whence  x^e^  =  — ,  and  differentiating  twice,  we 

U/  z 

1  I 

e"       e^  ~  -       e^  ' 

have  —  =  —==  00,  when  «  =  0.    .-.  y  =  e^sinx  =  xH"^  =  -  z=  cc,  when  x  =  0. 

Ex.  4.  Evaluate  y  =  ^■'"  log"  a?  for  a;  =  0. 

1 

,     „  nlog"— 'tC'- 

SuG.    y  =  — f—  =  — ,  when  a;  =  0.     Now  differentiating,  yx=o  = = 

1  00,  °  m 

x^  x'^+^ 

1         ,  w(n  —  l)log"-2a;--  ^   , 

-^— — —  (differentiating  again)  =  — — — — — ^—  =  -2^ '^    — .     After 

m  m2  m^ 


[n{n-  V,{n-2) 3  .  2  .  1]  i 

n  differentiations,  we  have  Vx=o  =  '  = 

±  — 

n{n-l){n-2) 3.2.1       v{n-l){n-2) 3.2.1 

[It  is  sometimes  expedient  to  put  the  function  in  the  form  —  rather  than  -. 

00  0 

Experiment  must  decide  which  is  preferable  in  any  given  case.  ] 


14:6,  Proh. — To  evaluate  y  =  f(x)  —  ^(x)  for  x  ==  a,  when  for 
this  value  of  the  variable  the  function  assumes  thefoi^m,  cc  —  cx). 

Solution. — Since /(x)  =  oo,  and  <p{x)  =  oo,  we  may  write  2/  =  — -—  = 

/(cc)         <p{x) 


mix)         fix)  0 

^-— — -— ^— ^ —  =  -.     Having  put  the  function  in  the  latter  form  it  may  he  treated 

/(X)  .  (p{X) 

as  in  the  first  case  {14:3). 

2               1 
Ex.  1.  Evaluate  y  = 7 r  for  x  =  1. 

2  1  1         a;2 1 

Solution. — ^In  this  case/(ic)  =  — - — r,  and  <z)(x)  = -.     Hence  -r-r  =  — — — , 

I  X  1  9  1 

and =  — - — .      "We  may   therefore    write    y 


q)[x)  1  x2  —  1  X  —  1 

X  —  1       x^  —  1  «-j-l 

~T~~~2~       ~T~      i~x      ^..^      ,.  ^.    .      —  1         1 

(differentiating)  =  — —  == 


x^  —  l    x  —  1  ^—  1         x2  _  i^^j  ^  &'        2a;  2" 

~2  r~    •  2 

[In  this    case  the   factor  1  —  x  can  be  divided  out  without    differentiating. 

2  1  2  — a;  — 1        1— x  1 

Moreover = 5 —  =  — 5-  = 7—-.] 

a;2  — 1      x  —  1         x'^  —  l         x^—1  x  +  l"* 


86  APPLICATIONS   OF   THE   DIFFERENTIAL   CALCULUS. 

X  1  1 

Ex.  2.  Evaluate  y  = —  , for  x  =  1.  Vx^i  =  ?:. 

X  —  1       log  X  2 

Ex.  3.  Evaluate  y  =  sec  x  —  tan  x  iov  x  =  — . 

SuG. — This  may  be  treated  exactly  as  the  last ;  but  the  following  is  more  ele- 

1  sin  X         1  —  sin  x         0       ,  tC 

gant.     V  =  sec  a;  —  tanx  = —  =  =  7:,   when  x  =  —. 

cosx         cosic  cos  a;  0  2 

Whence,  dijfferentiating,  y     ^  = : =  0,     Therefore  when  ^  =  — ,  secx  and 

r. — —  sm  X  * 

tan  X  are  equal,  a  fact  not  diflficult  to  observe  from  a  figure. 

1  X 

Ex.  4.  Evaluate  y  =  r for  a;  =  1.  y^^^  =  —  1. 

log  X       log  X 


147.  JProh.—To  evaluate  y  =  {{{■k)}'^^'^^  for  ^  =z  2^,  when  for  this 
value  of  the  variable  the  function  assumes  either  of  the  forms  0°,  c»°, 

. 

Solution. — Passing  to  logarithms  we  have  log  y  =  {cpx)  log/(x).  "When 
f{x)  =  0  and  (p[x}  =  0,  log  y=(p{x)  log/(x)x=a  =  0  X  ( —  00)  ;  when/(iK)  r=  00  and 
<p[x)  =  0,  log  2/  =  <p(a;)  log/(a)i=a  =  0  X  00;  when /(a;)  =  1  and  cp{x)  =  00, 
log  y  =  q){x)  log/(x)a-=o  =  00  X  0.    Hence  all  these  cases  fall  under  {14:5). 

Ex.  1.  Evaluate  y  =  x'  for  x  =  (i. 

Solution,    log  y  =  x  log  x  —     ^    —  — ,  when  a;  =  0.     Whence,  replacing 

X 

numerator  and  denominator  by  their  first   differential  coeflBcients,  we  have 
1_ 

■        = =  —  a;  =  0,  for  a;  =  0.     Hence,  log  2/  =  0,  and  y  =  \, 

1  X 

Ex.  2.  Evaluate  y  =  af' "^  and  y  =  (sina;)'"''  for  x  =  0. 

Sua.  — Since  for  x  =  0,  sin  x  =  x,  these  are  each  =  a;^  •* .  2/a;=o  =  (sin «)**"*  = 
xi'=l,  by  Ex.  1. 

These  may  also  be  solved  directly.     Thus  y  ==  a;^'°  ^,  gives  log  y  =  sin  x  log  x  = 


cosec  X            00 

1 

a  *"■— >  •'-  "- 

,                   logx 
logy.=o=-Y- 

X 

sin'x 

2  sin  X  cos  x 

2  sin  X  cos  x 

- 

COSiC 

iccosa; 

cosx  —  xsina; 

cosx 

sin  X 

sin2  X    ? 

2  sin  X  =  0.     .  • . 

y 

=  1. 

EVALUATION  OF   INDETERMINATE   EXPRESSIONS.  87 

,  .         .        .         1                  .       ,        .              loar  sin  x        , .  ^  ... 

Also  y  =  (smir)*'"==  gives  logy  =  sm a; log sm ^  =  —- (differentiating) 

COS6C  *Cx=  0 

cot  a;  1  . 

=  0.     .  • .  2/  =  1. 


cosec  X  cot  X  cosec  x 

Ex.  3.  Evaluate  y  =  (cot^)^*'^*  for  x  =  0. 

SuG. — Put  this  in  tlie  form  — .     Thus  logv  =  sin  x  log  cot  a;  =  — ■  (dif- 

00  ^  cosec  iCa:=0 

cosec2  a; 


-  .     .  cot  X  cosec  X         sm  cc         0         . 

lerentiatme)  = = = =  -  =  0. 

—  cosec  X  cot  X         cot^  x         cos^  x         1 


Ex.  4.  Evaluate  y  =  (1  +  nxy  for  x  =  0, 

„  .  log(l   4-  nx)  0      ^.^        1.  X-        1  5^ 

SuG.     log  ?/  =  ^^ ■ =   -.     Differentiating,  log  2/2=0  =  -   =  n. 

X  x  =  0  ^  •*• 

,  • .  y=e\ 
Ex.  5.  Evaluate  y  =  {Qosiax)}"""'"^'^^  for  ^  =  0. 


SuG.     y  =   {cos(ax)}<'°^«<=^^'=^)  =   1",    when    x  =  0.      Passing  to   logarithms 

log  cos  («x)         0 

s  (ax)   =  — ^ ^^ =  -,   when  x  =  0.     Differentiat- 

sin2  (cxj  0 

—  a  tan  (ax)  a  tan  (ax)       —  a-  sec2(ax)  a- 


mg  twice    ogT/^^o        2c  sin  (ex)  cos  (cic)  csin(2ca;)        2c2cos(2ca;)  2e^ 


«       * 


88  APPLICATIONS  OF  THE  DIFFERENTIAL  CALCULUS. 

SUCTION     III. 

Maxima  and  Minima  of  Functions  of  One  Variable, 

14:8,  Def. — A  M^aociniutn  value  of  a  function  of  a  single  vari- 
able is  a  value  which  is  greater  than  the  immediately  preceding  and 
the  immediately  succeeding  values  ;  i.  e.,  the  value  when  the  variable 
takes  an  infinitesimal  decrement,  and  the  value  when  the  variable 
takes  an  infinitesimal  increment. 

Ill's. — Let  y  =  sin  x.  When  x  =  —,  y  \b  b,  maximum,  since  it  is  greater  than 
the  immediately  preceding  and  the  immediately  succeeding  values.  If  x  takes  an 
increment  h,   making  y'  =  sinf^ f-  ^  )>  or  a  decrement,  —  h,  so  that  y"  = 

sinf  —  —  ^^\y  is  evidently  greater  than  y'  and  y" ,  as  at  90°  the  sine  is  greater 

than  it  is  at  a  little  more  or  a  little  less  than  90°. 

Again,  constructing  the  equation  y'^  =  603^  —  x'^,  we 
find  the  right  hand  branch  to  be  as  given  in  the  figure. 
Here  y  =:f{x),  and  y  is  &  maximum  when  a;  =  A  D  = 
4,  since  for  x  infinitesimally  less  or  greater  than  4,  y  is 
less  than  for  x  =  4.      The  maximum  value  of  y  is, 


therefore,  y  =  -^6-4^  — ^4^  =  3^  nearly.  ^xq   19 

Once  more,  let  y  =  8x  —  a;^.  If  x  =  1,  y  =  7  ',  if 
x  =  2,  2/  =  12  ;  if  a;  =  3,  2/  =  15  ;  if  x  =  4,  2/  =  16  (a  maximum) ;  if  cc  =  5,  ?/  =  15  ; 
if  a;  =  6,  2/  =  12  ;  and  if  a;  =  7,  y  =  7.  Hence  it  appears  that  as  x  increases  y  in- 
creases till  it  has  attained  a  certain  value,  when  although  x  is  made  to  continue 
its  increase,  y  begins  to  diminish.  The  point  at  which  the  function  ceases  to  in- 
crease and  begins  to  decrease  is  its  maximum.  In  this  case  it  will  be  found  that 
however  little  x  varies  from  4,  either  way,  y  becomes  less  than  16.  Thus  if  x  ^= 
3.9,  y  =  15.99  ;  and  if  a;  =  4.1,  y  =  15.99. 

149,  I^EF. — A.  ]\Hnil7lU7¥l  value  of  a  function  of  a  single  vari- 
able is  a  value  which  is  less  than  the  immediately  preceding  and  the 
immediately  succeeding  values  ;  i.  e. ,  the  value  when  the  variable 
takes  an  infinitesimal  decrement,  and  the  value  when  the  variable 
takes  an  infinitesimal  increment. 

tc 
Ill's. — Let  2/  =  coseca;.     As  x  approaches  —  y  diminishes  and  approaches  1, 

reaching  1  at  a;  =  -  .     "When  x  passes  — ,  y  begins  to  increase,  so  that  2/  =  1>  is  a 

it  A 

minimum  value  of  the  function  y  =  cosec  x. 

Again,  y  =  x"  —  Q>x  -\-  10,  has  a  minimum  value  for  a;  =  3,  at  which  value 


MAXIMA  AND  MINIMA  OF  FUNCTIONS  OF  ONE  VARIABLE. 


89 


y  =  1.  By  substituting  values  of  «  a  little  greater  than  3, 
as  3.01,  and  a  little  less  as  2.09,  y  will  be  found  to  be  greater 
than  1  in  both  cases.  The  locus  of  the  function  is  given  in 
Mg.  20,  where  P  D  represents  the  minimum  value  of  y. 

ISO,  CoE. — The  same  function  may  have  several 
maxima  or  several  minima  values,  and  these  may  he 
equal  or  unequal.  Moreover,  a  maximum  value  may 
be  equal  to  or  even  less  than  a  minimum  value  of  the 
same  function. 

Ill's.— The  function  y  :=  x^  —  8x3  _^  22x2  —  24a;  -}-  12,  has 
minima  values  for  x  =  1,  and  x  =  3,  which  values  are  both 
?/  =  3  ;  or  two  equal  minima  values,  as  illustrated  by  the  ordi- 
nates  at  P  and  P"  in  the  figure.  For  x  =^2  y  =  i,,  a  maximum 
value,  as  illustrated  by  P'D'. 

Again,  let  y  = /(.t?)  be  the  equation  of  M  N  referred  to  AX 
and  AY  Fig.  22.  Then  PD,  P"D",  and  P"'D'"^are  maxima 
values  of  y ;  and  P'D',  and  P"'D"'  are  minima  values. 
But  the  several  maxima  values  are  unequal  and  the  minimum 
P'  D'  is  greater  than  the  maximum  P^vQiv. 


Fig.  20. 


ADD'  D" 

Fig.  21. 


151,  ScH. — It  will  be  observed 
that  the  terms  maximum  and  mini- 
mum, as  here  used,  do  not  mean  the 
greatest  possible  and  least  possible. 
Thus,  if  we  ask  for  the  maximum 
value  of  3/  in  3/  =  «;•*  —  Zax'^  —  5,  we 
do  not  inquire,  "what  is  the  greatest 
possible  value  which  y  can  have  ? 
but  simply,  whether  if  x  vary  con- 
tinuously through  all  possible  values, 
there  is  any  point  at  which  y  will  at- 
tain a  greater  value  than  it  had  immediately  preceding  that  point,  and  than, 
it  will  have  immediately  after  passing  that  point ;  and,  if  there  be  such  a 
value  of  y,  what  it  is. 


Fig.  22. 


1S2,  JPfop. — In  an  explicit  function  of  a  single  variable,  y  =  f(x), 

dy 

the  first  diffey^ential  coefficient,  —jChanges  sign  from,  -f  to  — ,for  contin- 
uously increasing  values  of  the  variable,  where  the  function  is  at  a  maxi- 
mum, and  from  —  to  -{-  where  the  function  is  at  a  minimum.  Hence  for 
such  values  the  first  differential  coefficient  ==  0  or  oo. 

Dem. — Let  y  =f^x)  be  the  function.     First,  For  x  =  x',  suppose  y  becomes  y' ,  a 
maximum.     Then  ?/'  =  f'^x')  is  at  a  maximum.     Now  the  immediately  preceding 


90 


APPLICATIONS   OF  THE   DIFFERENTIAL  CALCULUS. 

dx')  —fix') 


flu*  "fi  % 

state  of  tlie  function  is  fix'  —  dx"),  and  we  have  -4-,  =  — — ; tt- 

''^  "  dx  {x'—dx) 


By 


hypothesis /(a'  —dx)  — f{x')  is  — *,  and  as  (x'  —  dx')  —  x'  is  evidently  — ,  we  have 

dv' 

—  -j—     Again,  the  immediately  succeeding  state  to  y'  =f{x')  isf{x'-\-dx'); 


hence  we  have  -r—,  = 
dx 


dy'       f{x'-\-dx')—f{x') 


\x'  -j-  dx)  —  x 
and  as  {x  -\-  dx)  —  x    is  evidently  -\-,  we  have 


By  hypothesis /(a'  -\-dx')  — /(.'r')  is 
dy' 


dx 


Therefore  where  y'  = 


dy' 


fyX  )  is  a  maximum  -—  changes  sign  from 


to 


dx')-f{x') 


dx  )  —  x 

x'    is  evidentlv  — 


is  — ,  since  by 
Again 


dv'       fix'  ■ 
Second.  If  y'  =  fix')  is  at  a  minimum  -^^  =  — — - 

dx  (X 

hypothesis  fix  —  dx')  — fix')  is  -|-,  and  (cc'  —  dx') 

dy'        f(x'  +  dx')  —fix)  .  ^-u     •    ^-  '    ,    ^  'N       ^/   'N  •      I         A 

d^'  ^     ix   +  dx)  —  x'         "^'  "'^  ^  hypothesis  /,a:    +  dx  )  —fix  )  is  +,  and 

{x   -\-  dx)  —  x  is  evidently  -j-- 

Finally,  since  when  a  varying  function  changes  sign  it  passes  through  0  or  go, 

dy 
we  have  -^  =  0  or  oo  for  maxima  and  minima  values  of  the  function,     o.  e.  r». 
dx 

Geometeical    IiiLUSTEATioN.  —  T"S     being 

tangent  to  the  curve  M  N  at  P,  P'  being  a 

consecutive  point  so  that  P  E  represents  dx, 

and    P'E   dy,   we   observe    that    the    angle 

P'PE  =  a,  the  angle  which  the  line  makes 

with  the  axis  of  abscissas.     Hence  tan  a  = 


tan  P'PE  = 


-=r-r=-  =  -  -  ;  i.  e.  the  first  dif- 
PE         dx 


Fig.  23. 


ferential  coefficient  of  the  ordinate  regarded 

as  a  function  of  the  abscissa,  represents  the  tangent  of  the  angle  which  a  tangent 

to  a  plane  curve  makes  with  the  axis  of  abscissas. 

Now,   observing  Fig.  22  we  see  that  as  x  is  increasing,  and  y  approaching  a 
maximum  value  as  PD,  the  tangent  to  the  curve  makes  an  acute  angle  ;  hence 


dy 
approaching  P  from  the  left  --  is 

dx 


At  P  the  tangent  becomes  parallel  to  the 


axis  of  X  ;  tan  a 


d'd^ 

--  =  0.     Immediately  upon  passing  P,  a  becomes  obtuse, 

and  consequently  tan  a  =  -^is  — . 

dx 

So  also  in  approaching  a  minimum  value  as  P'  D'  from  the  left  it  appears  that 

a  is  obtuse,  and  hence  -- 
ax 


passing  P',  a  becomes  acute  and  --  4-. 

dx 


;  at  this  point,    P',  a  =  0,  and 
dy 


dy 

dx 


0;  and  after 


*  The  hypothesis  is  that  y'  =J\x')  is  a  maximum,  i.  e.  is  greater  than  either  the  immediately 
preceding  or  the  immediately  succeeding  state  of  the  function.  But  fix'  —  dx')  is  the  immedi- 
ately preceding  state,  and/(a;'  +  dx')  is  the  immediately  succeeding  state.  Hence/(a;''—  flfa;0< 
/(xO,  and/(a3' +  cfaK)  </(a:0. 


MAXIMA  AND  MINIMA   OF  FUNCTIONS   OF  ONE  VARIABLE.  91 

rh%t 

To  illustrate  the  case  in  which  -;-  changes  sign     ^ 

by  passing  through  oo,  consider  y  =/(.r)  as  the 
equation  of  M  N ,  tig.  24.  P  D  is  evidently  a 
maximum    ordinate.      But  in   approaching    PD 

from  the    left,    a  is  an  acute  angle,  and  --,  -}-• 


D  X 

At  P,  a  =  90O,  and  j(=  oo.     After  passing  PD,  ^^^-  2^' 

dv 
a  is  obtuse  and  --,  — .     A  similar  illustration  may  be  given  of  the  case  in  which 

<^V  .1  -.  ,         .   . 

■J-  passes  through  go  at  a  minimum. 

ScH. — The  student  needs  to  guard  against  the  error  of  supposing  that  all 
values  of  the  variable  which  render  the  first  differential  coefficient  0  or  oo, 
necessarily  render  the  function  a  maximum  or  minimum.  These  values  of 
the  variable  correspond  to  the  maxima  and  minima  values  of  the  function  if 
it  has  any  maxima  or  minima  values,  since  if  the  first  differential  coefficient 
changes  sign,  it  must  pass  through  0  or  oo  ;  but  a  quantity  may  pass  through 
0  or  00  without  changing  sign,  so  that  the  values  of  the  variable  which 
render  the  first  differential  coefficient  0  or  oo  are  simply  critical  values,  i.  e. 
values  to  be  examined. 

153.  JPvop, — In  an  explicit  function  of  a  single  variable,  y  =  f(x), 
the  second  differential  coefficient,  —^,  if  not  0,  is  —  where  the  function 
is  at  a  maximuin,  and  +  where  it  is  at  a  minimum. 

Dem. — Let  y  =/(x)  be  the  function.     We  have  seen  that  when  the  function 

dv 
passes  through  a  maximum  -~  changes   sign  from  -|-  to  —  for  continuously  in- 

dx 

dy 
creasing  values  of  x,  i.  e.  --  is  decreasing  ;  and  when  the  function  passes  through 

a  minimum  --  changes  sign  from  —  to  +,   i.  e.  --  is  increasing.     jNow  -—  = 

ClX  CIX  CtX" 

d^y 

dx  df'{x)        f  (x -\- dx) — fix)      ,.  ,    .  V       xu  J.      •  J 

==   -^-— ^ —  =  '^—^ ! '—-—,  which  IS  —  when  the  numerator  is  — ,  and 

dx  dx  dx 

dy 

-\-  when  the  numerator  is  -f- ,  since  dx  is  -J-  by  hypothesis.     But  at  a  maximum  -- 

is  decreasing  for  increasing  values  of  x,  and  f{x  -f-  dx)  — f{oc)  is  —  ;  and  at  a 

dv 
minimum  --  is  increasing  for  increasing  values  of  x,  and /'(a;  +  dx)  — f{x)  is  -\- . 

dHi 
Therefore  -r^  is  —  at  a  maximum  value  of  the  function  and  4-  at  a  minimum, 

unless  it  is  0,  a  case  which  is  not  yet  provided  for.  • 

154,  ScH.  1. — The  ordinary  method  of  examining  an  explicit  function 


92  APPLICATIONS  OF  THE  DIFFERENTIAL  CALCULUS. 

of  a  single  variable  for  maxima  and  minima  values  is  to  form  the  first  differ- 
ential coefficient,  put  it  equal  to  0,  and  solve  the  resulting  equation.  Some  or 
all  of  the  values  of  the  variable  thus  foulid  may  correspond  to  maxima  and, 
minima  values  of  the  function.  They  are  then  to  be  examined  separately. 
To  do  this,  form  the  second  differential  coefficient  of  the  function  and  sub- 
stituting in  it  the  value  of  the  variable  to  be  examined,  if  it  gives  a  —  re- 
sult, this  value  of  the  variable  corresponds  to  a  maximum  value  of  the 
function  ;  but  if  it  gives  a  -|-  result,  it  corresponds  to  a  minimum  vahie  of 
the  function.  Thus  all  the  values  of  the  variable  arising  from  equating  the 
first  differential  coefficient  with  0,  are  to  be  examined.  Tf,  however,  any  one 
of  these  critical  values  renders  the  second  differential  coefficient  0,  it  is  best 
to  examine  the  first  differential  coefficient  for  this  value  and  see  if  it  actu- 
ally does  change  sign  in  jiassing  from  a  value  of  the  variable  infinitesimallj 
less  to  a  value  infinitesimally  greater  than  that  being  examined. 

155,  ScH.  2. — The  following  axiomatic  principles  often  facilitate  the  ex- 
amination of  a  function  for  maxima  and  minima  values  : 

1st.  Whatever  value  of  x  renders  u  =  f[x)  a  maximum  or  minimum,  ren- 

ders  u'  =  afix)  or  u"  =  —-!■  a  maximum  or  minimum.     Hence  constant  fac- 

a 

tors  or  divisors  may  be  dropped  from  the  function. 

2nd.  Whatever  value  of  x  renders  u  =/{x)  positive  and  a  maximum  or 
minimum,  renders  u  =  [/(•'c)]"  a  maximum  or  minimum,  h  being  a  positive 
integer;  but  if  u=f{x)  is  rendered  negative  for  the  particular  value  of  x, 
u  =  [f{x)y"'  is  a  minimum  when  it  =f[x)  is  a  maximum,  and  a  maximum 
when  u  =f{x)  is  a  minimum.  Hence  the  function  may  be  involved  to  any 
power. 

3rd.  Whatever  value  of  x  renders  u  =  log  [/(a;)]  a  maximum  or  minimum 
renders  u'  =  f(x)  a  maximum  or  minimum.  Hence  to  examine  the  log- 
arithm of  a  function  we  have  to  examine  simply  the  function  itself  dropping 
the  symbol  log. 

Ex.  1.  What  values  of  x  render  y  =  v  4a-^-  — ^  "lax;^  a  maximum  or 
minimum  ;  and  what  are  the  maxima  and  minima  values  oi  y? 


Solution.* — Whatever  value    of  x  renders  y  =  s/ia  x-  —  'Aiw-'  a  maximum  or 

minimum  rendei"S  y^  or  y'  =  -ia'x-  —  2ax^  a  maximum  or  minimum  {155).     And 

for  a  similar  reason  we  maj'  drop  the  constant  factor  2a,  and  examine  y"='2ax'^  —  x^, 

since  any  value  of  x  which  renders  the  original  function  a  maximum  or  minimum 

dy" 
will  also  render  this  a  maximum  or  minimum.     Differentiating  we  have  -^  = 

4ax  —  3x2.     Now  whatever  A^alue  of  x  renders  the  function  a  maximum  or  mini- 

4a 
mum  renders  4ax  —  3x2  =  0.     From  this  x  =:  0,  .r  =  — .     If,  therefore,  there  are 

o 

any  maxima  or  minima  values  of  the  function,  they  are  those  which  correspond  to 

*  This  solution  may  seem  needlessly  prolix,  but  the  author  finds  that  comparatively  few  stu- 
dents really  follow  the  argument  through  unless  required  to  give  it  thus  in  dt-tail. 


MAXIMA  AND   MINIMA   OF   FUNCTIONS   OF   ONE   VARIABLE.  93 

one  or  the  other  or  both  of  these  vahies  of  x.     Diiferentiating  again,  —4  =  4a  —  6cc. 

For  X  =0,  — ^  =  4^a  ;  hence  a;  =  0  corresponds  to  a  minimum  value  of  the  func- 
dx'^ 

tion.     For  x  =  — ,  —  =  4a  —  8a  =  —  4a  ;  hence  x  =  --  corresponds  to  a  maX' 
3     dx'  o 

imum  value  of  the  function. 


Substituting  these  values  of  x  we  find  y  =  v/4a'av^  —  2ax-  =  0,   a  minimum 

6i«*       128«'        8a2 


value  ;  and  y  =  >/4a2ic2  —  'lax:^  =       —r- ^r^r-  = -,  a  maximum. 

\    y  ^'  3\/3 

Ex.  2.  What  values  of  x  render  y  =  x-^  —  9^=^  +  24.x  —  16  a  maxi- 
mum or  a  minimum,  and  what  are  the  maxima  values  of  y  ? 

Results,  X  =  2  corresponds  to  a  maximum,  and  x  =  4:  to  a  mini- 
mum.    The  maximum  value  is  y  =  4,  and  the  minimum  y  =  0. 

Ex.  3.  Examine  y  =  x^  —  ^x^-  —  24.x  +  85  for  maxima  and  minima. 
Results,  For  ^  =  4,  ?/  =  5,  a  minimum  ; 

For  X  =  —  2,  y  =  113,  a  maximum.    , 

Ex.  4.  Examine  y  =  5(x  —  ^r^)  for  maxima  and  minima. 
SuG. — Drop  the  5.     x  =  i,  gives  y  ==.  ^,  a  maximum. 

Ex.  5.  Examine  y  =  {2ax  —  x^)'^  for  maxima  and  minima. 

SuG.  — Use  y'  =  2ax  —  x^.     x  =  a,  gives  y  =  a,  a  maximum,  and  —  a,  a  min- 
imum. 

Ex.  6.  Examine  y  =  x*  —  Sx^  -\-  22a;2  —  24^  +  12  for  maxima  and 
minima. 

Sug's.     ^f  =  4x'^  —  24.'k2  4-  44a;  —  24  =-.  0,  or  x^  —  Gx2  +  IL-r  —  G  =  0.     To 
ax 

find  the  roots  of  this  equation,  observe  that  the  factors  of  the  absolute  term  with 

its  sign  changed  are  1,  2,  and  3  (Complete  School  Algebra,  111).     By  trial 

these  are  found  to  be  the  values  oi  x,  x  =  l  gives  ?/  =  3,  a  minimum  ;  .^•  =  2  gives 

2/  =  4,  a  maximum  ;  .r  =  3  gives  ?/  =  3,  a  minimum  (see  III.  Fig.  21). 

Ex.  7.   Examine  y=x^  —  5x^  +  5x^  +  1  foi'  maxima  and  minima. 

Results,  The  critical  values  of  x  are  0,  0,  1,  3.  For  x  ■=  1,  ij  =  2, 
a  maximum  ;  for  x  =  d,  y  =  — 26,  a  minimum,  x  =  0  does 
not  correspond  to  either  a  maximum  or  minimum  value  of  y. 

SuG. — That  x  =  0  does  not  correspond  to  either  a  maximum  or  a  minimum  is 

determined  as  follows  : 

dy 
Having  --  =  5x4  —  20x3  +  15x2,  substitute  0  —  h  and  0  -\-  h  for  x,  and  evaluatef^ 
dx 

dv 
the  expression  for  k  infinitesimal,  thus  determining  whether  --  changes  sign  or  not^ 

cvX 


94  APPLICATIONS   OF   THE  DIFFERENTIAL  CALCULUS, 

in  passing  through  x  =  0.     Thus  JJ  =  5(0  —  h)"  —  20(0  —  hy  -f  15(0  —  A)2  = 

5;i4  +  20/13  +  157^2  =  15;i2,  when  h  is  infinitesimal.    Again  --    =  5/i4  —  20A3 

ux- 

-f-  15^2  ^=  15^2,  when  h  is   infinitesimal.      Therefore,  as   --  has  like  signs  on 

both  sides  of  a;  =  0,  and  consecutive  with  it,  it  does  not  change  sign  in  passing 
through  a;  =  0.  Hence  jb  =  0  does  not  correspond  to  either  a  maximum  or  a  min- 
imum. 

Ex.  8.  Examine  y  z=h  -\-  \x  —  a)^  for  maxima  and  minima. 

dv 
Stjg's.      —  =  3(a;  —  aY  =  0,  gives  x  =  a.     Hence  if  there  is  any  maximum  or 

dv 
minimum  it  must  be  v  =  &,  as  no  other  value  of  x  than  x  =  a  will  render  -^  =  0. 

dx 

d^V  dv 

Again,  since  this  value  renders  -r—  =  0,  we  examine  it  by  ascertaining  whether  -rr 

dx^  dx 

dv  dv 

changes  sign  a.tx  =  a.     -^  =  3(a  —  h  —  a)2  =  3h'  is  the  value  of  -^  next  preced- 

(XX  ax 

dv 
ing  X  =  a;  and  —■  =  3(a  -f-  ^  —  a)^  =  STi^  is  the  next  succeeding  value.     There- 

fore,  as  --  does  not  change  sign  at  a;  =  a,  the  function  has  no  maximum  nor  mini- 
dx 

mum  value. 

Ex.  9.  Examine  y  =  a{x  —  by  -\-  c  for  maxima  and  minima  values. 

Sug's.     -^  =  4:(x  —  hy  =  0.     .-.  x  =  h.     %  =  4(&  —  h  —  hy  =  —  4.h^,  and 
dx         ^  '  dx 

dv'  dv' 

-J—  =  4(6  4"  ^  —  6)3  =  47i3,  are  the  values  of  — —  immediately  preceding  and  suc- 

(X*o  CuC 

dy' 
ceedmg  x  =  h  ;  hence,  as  — —  changes  sign  from  —  to  4-  ^t  this  point,  x  =  6  cor- 

dx 

responds  to  a  minimum.     .-.  y   =  a(b  —  6)^  -J-  c  =  c  is  a  minimum. 

Ex.  10.  Examine  y  =  {x  —  l)^{x  -\-2y  for  maxima  and  minima. 

Sug's.     -^  =  A{x  —  iy{x  +  2)3  -f-  3[x  —  iy{x  -f  2)2  ==  {{x  —  l)3(x  +  2)-'} 

{4(a;-f2)4-3(a;  — 1)}  =(«  — l)3(a;+2)2(7a;-f5)=0.  .-.  .r— 1  =  0,  a;+2  =  0, 
7a;  -|-  5  =  0,  give  x  =  1,  x  :=  —  2,  x  =  —  fas  the  critical  values  of  x. 

d^V 

-^=.3(a;  — l)2(a;-|-2)2(7x  +  5)4-2(x  — l)3(x  +  2)(7a;  +  5)-l-7(a;  — l)3(cc  +  2)2  =  0 

(123)  .  92 
for  X  =^1,  and  x  =  —  2,  but  is  —  -^ — —- for  x  =^  —  ^.     The  latter  value,  there- 

'  74  ' 

12    9' 
fore,  corresponds  to  a  maximum,  and  gives  y  =z  ( — f —  ly{ — ^  -|-  2)3  =:  — — — ,  a 

maximum. 

To  ascertain  whether  x  =  \  corresponds  to  a  maximum  or  minimum,  notice 


MAXIMA   AND    MINIMA   OF   FUNCTIONS   OF  ONE   VARIABLE.  95 

that  -^  =  n—h~  1)3(1  —  h-^  2)2(7  —  7h-{-5)=  —  h^S  —  /i)2(12  —  Ih)  is  - 
ax 


and  '^^  =  a+h-  1)3(1  +  /i  +  2)2(7  +  7/i  +  5)  =  h\3  +  70^(12  +  7h)  is  +. 

Hence  at  x  =  1,  --  changes  sign  from  —  to  +,  and  there  is  a  minimum  at  this  value. 
dx 


dy 

dy_ 
dx 
This  minimum  is  2/  =  0. 

Finally,  to  test  a;  =—  2,^  =  (—  2  —  /i  — 1)3(—  2  —  /i  + 1)\— 14  —  7/i  +  5)  = 

(—  3  —  K)\-~  h)\—  9   —  7/i),  which  is  -f .      Again,  ^  ^  (—  2  +  /i  —  1)^ 

(_  2  4-  /i  +  2)2(—  14  -I-  7/i  4-  5)  =  (—  3  -f  /i;3(+  )i)\—  9  +  7/i)  is  also  +. 
Therefore  a;  =  —  2  does  not  correspond  either  to  a  maximum  or  a  minimum. 

156.  ScH. — It  is  usually  easy  to  see,  without  going  through  with  the  de- 
tails of  the  substitution,  whether  -^  changes  sign  with  h  in  such  cases  as 

dx 

dy 
the  above  ;  that  is,  whether  \i  x=  a  is  the  critical  value  we  are  testing,  -~ 

will  have  a  different  sign  when  we  substitute  a  -\- h,  iox  x,  from  what  it  will 
when  we  substitute  a  —  %  for  x. 

Ex.  11.  Examine  v  =  -^ —     J,    for  maxima  and  minima. 

^        (a;  —  3)2 

Stjg's.     --  =  -  =  0,  gives  for  the  critical  values  re  =  —  2,  x  =  13. 

d«  (a?  —  3)3 

dy  _  (x  +  2)2(c.-13)  _  ^  _  3     _      ^j^^^^^  ^  _  3_ 

da;  (X  —  3)3  .  5         V  ^ 

d^v 
In  this  case  it  is  better  not  to  form  -^  as  it  is  complicated,  but  test  the  critical 

dx"^ 

dv 
values  by  noticing  whether  --  changes   sign  or  not  for  these  points,     x  =  —  2 

does  not  correspond  to  either  a  maximum  or  a  minimum,     x  =  13,  gives  :y  =  33|, 
a  minimum,     x  =  3,  gives  i/  =  oo,  a  maximum.  ^ 

ScH.— The  first  10  examples  give  x  =  co  iov  -j  =  oo,  and  hence  give  rise 

to  no  critical  values,  as  .r  =  oo  cannot  correspond  to  a  maximum  or  mini- 
mum, there  being  no  succeeding  value  of  the  function. 

{X  1)2 

Ex.  12.  Examine  y  ==  -, =^-  for  maxima  and  minima. 

{x-\-iy 

Sug's.— Putting-5-  =  0,  gives  x  =  1,  and  5,  as  the  critical  values.     Putting 

—  ==  oc,  gives  X  =  —  1.     When  x  =  1,  2/  =  0,  a  minimum.     When  x  —  5,  y  =  j^y, 

dx 

a  maximum.     When  x  =.  —  1,  y  is  neither  a  maximum  nor  a  minimum. 


96  APPLICATIONS   OF  THE  DIFFERENTIA!,   CALCULUS. 


3 


Ex.  13.  Examine  y  =  b  -{-  {x  —  aj"^  for  maxima  and  minima. 

Result,  The  critical  value  oi  x  i^  x  =  a.     But  this  does  not  corres- 

dy 
pond  to  either  a  maximum  or  a  minimum,  since  -^  does  not 

change  sign  at  this  value. 

m 

SuG.  —In  this  example  — ^  =  rb  ex  for  a;  —  a,  and  hence  cannot  be  used  to  dis- 
criminate  between  maxima  and  minima. 

4 

Ex.  14.  Examine  y  =  h  -{-  {x  —  a)^  for  maxima  and  minima. 

Result,  ?/  =  6  is  a  minimum. 

Ex.  15.  Examine  y  =  i  —  {x  —  aj^  for  maxima  and  minima. 

Result,  2/  =  6  is  a  maximum. 

Ex.  16.  Show  that  y  =  x^  —  Sx-  +  6^+7  has  neither  a  maximum 
nor  a  minimum  value. 

X 

Ex.  17.  Show  that  y  =  - is  a  maximum  when  x  =  cos  a;.* 

1  4- j;  tana? 

1-  '' 


dv  cos"^  X        .__.  dy 

SuG.     —  =  —————  .     When  x  <"  cos  a;,   --  is  +  ;   but  when  x  >>  cos  a, 
dx       (l  +  a;tanx)2  ^  '   dx       '  ^ 


^is 
dx 


Ex.  18.  Show  that  y  =  sin^  x  cos  x  is  sl  maximum  when  x  =  60°. 

sin  X 

Ex.  19.  Show  that  y  = is  a  maximum  when  x  ^  4:5°. 

^        1  +  tan^ 


GEOMETRICAL   PROBLEMS. 

Ex.  1.  Eequired  the  altitude  of  the  maximum  cylinder  which  can 
be  inscribed  in  a  given  right  cone  with  a  circular  base. 

Solution. — Let  SO  =  a  be  the  altitude,  and  AO  =  6  the  radius  of  the  base 
of  the  given  cone.  Let  ac  =  xhe  the  altitude,  and  cO  =  of  =  y  he  the  radius 
of  the  base  of  the  required  cylinder.  The  function  which  is  to  he  a  maximum  is  the 
volume  of  the  cylinder.  Calhng  this  V,  we  have  V  =  Tty^x.  In  this  form  V  is  a 
function  of  two  variables  x  and  y.  But  these  variables  being  dependent  upon 
each  other,  we  can  find  the  value  of  one  in  terms  of  the  other.     Thus,  S/  :  S  O  : : 

«/  :  AO  ;  or,  in  the  notation,  a  —  x  :  a  : :  y  :h  ;   whence  y  =i  -[a  —  x).     Substi- 


*  When  X  =  cost,  x  =  42"*  21'  nearlj'. 


GEOMETRICAL  PEOBLEMS. 


97 


tutmg  this  value  of  y,  we  have  V  =  — (a  —  x^x, 
which  is  to  be  a  maximum.  Dropping  the  con- 
stant  factor  —  {153,  1st),  we  have  V'=  {a  —  x^x 


a^x  —  2ax-^-j-x^. 


dV' 
dx 


=  a"  —  4aa;  +  3x2  _  o  ; 


whence  a;  =  ia  ;  that  is,  the  axis  of  the  cyHnder  is 
i  of  the  axis  of  the  cone.  From  this  we  readily 
find  y,  the  radius  of  the  base  of  the  cylinder  =|&. 
.  • .  volume  of  cylinder  =  -^^ita}fi.  But  volume  of  A 
cone  =:  \Tidb~ ;  whence  volume  of  cylinder  =  f 
volume  of  cone. 


Fig.  25. 


Ex.  2.  To  find  the  axis  of  the  maximum  cone  which  can  be  inscribed 
in  a  given  sphere. 


771 


Sug's. — Let  AmB  be  the  semicircle  which,  re- 
volved upon  A  B,  generates  the  sphere,  and  Aa&  the 
triangle  which  generates  the  cone.  Let  AO=r, 
A&=::ic,  anda&=:2/-    Then  V  =  inr2/2x=i7rx2(2r — a), 

—2 

since  a6  =2/^  =  A6  X6B=£c(2r  —  x).  .•.a;  =  |r, 
or  the  altitude  of  the  cone  is  f  of  the  diameter  of 
the  sphere.  Volume  of  sphere  =  ^itr'^,  volume  of 
maximum  cone  =  -/f  X  a^rr^  ;  or  the  cone  :==:  -^^  of  the  sphere. 


ScH.— In  attempting  the  solution  of  such  problems,  first  notice  wliat  the 
function  is  which  is  to  be  a  maximum.  Thus,  in  Fx.  1,  it  is  the  volume  of 
a  cylinder ;  in  Ex.  2,  it  is  the  volume  of  a  cone.  Having  obtained  the  equa- 
tion expressing  the  function  in  terms  of  the  variable  or  variables  on  which 
it  depends,  if  there  are  two  dependent  variables  involved,  find  from  the 
conditions  of  the  problem  the  relation  between  these  variables,  and  sub- 
stitute for  one  of  them  its  value  in  terms  of  the  other.  Finally,  we  have 
a  function  of  a  single  variable,  which  can  be  examined  for  maxima  and 
minima  values  in  the  usual  way. 

Ex.  3.  Required  the  cylinder  of  greatest  convex  surface  which  can 
be  inscribed  in  a  given  right  cone  with  a  circular  base. 

SuG. — The  function  is  the  convex  surface  of  a  cylinder.  Using  the  same  notation 
as  in   Ex.   1,   and  letting   S   represent  the  function,   we  have    S  =  27tyx  = 

(a  —  x)x.     .' .  X  =-la,  and  S  =  -3—  ;  that  is,  the  altitude  of  the  cylinder  is  i 

that  of  the  cone  ;  and  the  convex  surface  of  the  cylinder  is  to  the  convex  surface 


of  the  cone  as  -  ;  s/a^  -j-  ^^j  o^  ^s  i  the  altitude  of  the  cone  is  to  its  slant  height. 


Ex.  4.  Required  the  maximum  Cylinder  which  can  be  cut  from  a 


98  APPLICATIONS   OF   THE   DIFFERENTIAL  CALCULUS. 

given  fephcre.     The  axis  of  the  cylinder  =  f  \/3  times  radius  of  sphere. 

The  cylinder  is  to  the  sphere  as  1 :  y'S. 

Ex.  5.  Eequired  the  area  of  the  greatest  rectangle  which  can  be 
inscribed  in  a  given  circle. 

The  rectangle  is  a  square,  and  its  area  =  2r2. 

Ex.  6.  "What  is  the  altitude  of  the  maxi- 
mum rectangle  which  can  be  inscribed  in  a 
given  parabola  ? 

Sug's. — Let  ac  =  X,  af  =  y,  and  AX  =  «.  Let 
A  be  the  function,  the  area  of  the  rectangle.  Then 
A  =  2a'?/.  From  the  equation  of  the  parabola 
aj"  =  2p  X  A/,  or  y^  =  2p{a  —  x)  ;  whence  A  = 
2x\/2p^a  —  X).     A'  =^  ax^  —  x^,  and  x  =  |a. 

Ex.  7.  Bequired  the  axis  of  the  cone  of  maximum  convex  surface 
which  can  be  inscribed  in  a  given  sphere. 

The  axis  ==  ^  the  radius  of  the  sphere. 

Ex.  8.  Required  the  altitude  of  the  maximum  cone  which  can  be 
inscribed  in  a  given  paraboloid,  the  vertex 
of  the  cone  being  at  the  intersection  of  the 
axis  of  the  paraboloid  with  the  base. 

Sug's. — Let  ABC  be  the  parabola  whose  revo- 
lution about  AS  as  an  axis  generates  the  parabo- 
loid. Let  AS  ==^  a  the  axis  of  the  paraboloid, 
oS  =  X,  the  altitude  of  the  cone,  and  ao  =  y  the 
radius  of  the  base  of  the  cone.  The  result  is 
X  =  ia. 

Ex.  9.  Required  the  maximum  para- 
bola which  can  be  cut  from  a  right  cone 
With  a  circular  base,  knowing  that  the 
area  of  a  parabola  whose  limiting  co-ordi- 
nates are  x  and  y  is  |-.r?/. 

Sug's. — LetSO=a,  BO  =  &,  AX=a;,and 

f  X  =  t/.     The  function  is  A  (the  area)  =  ^xy. 

But  aX  =  y  =  v^BX   X   XC  ;    and  CX   : 


Fig.  28. 


C  B  :  AX  :  S  B,  or  CX  :  26  :  :  a;  :  v/a'^  -f  62  ; 
2hx 


whence  CX  =  — p,  letting  S  =  Va'^  +  ^^  ^o^ 


brevity.     Then  BX 


CB  — CX 


25 


26x 


Fig.  29. 


GEOMETRICAL   PROBLEMS. 


99 


-> 


x).     Finally,  A  =  ^x 


m 


86 


-x{S  —  x)  ■=  --\/x\S  —  X),  and  A'  =  Sx^ 


—  X*.     The  result  is  a;  =  |/S^,  that  is,  the  axis  of  the  parabola  is  4  the  slant  height 
of  the  cone.     The  area  of  the  parabola  =  ibS\/3.     Notice  that  CX  =  |CB. 

Ex.  10.  From  a  given  quantity  of  material  a  cylindrical  vessel  with 
circular  base  and  open  top  is  to  be  made,  so  as  to  contain  the  greatest 
amount.     What  must  be  its  proportions  ? 

Sug's. — Let  X  =  the  altitude,  y  the  radius  of  the  base,   and   V  the  volume. 

dV  dy 

Then   V  =  Tty'^x  is  to  be  a  maximum.     Hence   -y-  =  S^/x-^  -\-  y'^  z=  Q,  or  y  = 


—  2a;--.     But  iTtyx  -f  7ty^ 

litydy  =  0  ;  whence  --  =  ■ 
dx 


dx  dx 

s,  the  surface.     Differentiating  ^itxdy  -f-  27tydx  -f- 

y 


Substituting,   y  =  — -— .     .  • .  y  =  x,  that 
x  +  y_  ^'  ^       x  +  y 


is,  the  altitude  =  the  radius  of  the  base. 


The  altitude  =       — • 


Ex.  11.  Of  all  right  cones  of  a  given  convex  surface  to  determine 
that  whose  solidity  is  the  greatest. 

The  altitude  =  \/2  into  the  radius  of  the  base. 

Ex.  12.  To  find  the  maximum  rectangle  inscribed  in  a  given  ellipse. 

Sug's.     A  =  4xy.     A'  =  xy.     — r—  r=  ?/  +  ^'y  "^  ^'     '^^y^ 


-f  ^2x2    —    ^2^2. 


dy 
dx 


dx 
B'^x 
A^y 


y  = 


dx 

dy        B^x'i 

X—  =  . 

dx         A^y 


X  '.y  ::  A:  B.     That  is,  the  sides  of  the  rectangle  are  to  each 
other  as  the  axes  of  the  ellipse.     The  sides  of  the  rectangle 

are  A\/2,  and  B\/2. 


Fig.  30. 


Ex.  13.  To  find  the  maximum  cylinder  which  can  be  inscribed  in  a 

given  ellipsoid,  generated  by  the  revolution  of  an  ellipse  about  its 

2 
The  axis  of  the  cylinder  =  — -=A. 


transverse  axis. 


Ex.  14.  A  person  being  in  a  boat  3  miles  from  the  nearest  point  of 
the  beach,  wishes  to  reach  in  the  shortest  time  a  place  6  miles  from 
that  point  along  the  shore  ;  supposing  he  can  walk  5  miles  an  hour, 
but  pull  only  at  the  rate  of  4  miles  an  hour, 
required  the  place  where  he  must  land. 

Sug's. — Let  AX  =;r,  and  f  =  the  time  required  to 
reach  A  by  rowing  from  B  to  X,  and  walking  from 


5  4 


is  to  be  a  mini- 


mum.   He  must  land  at  X,  1  mile  from  A. 


FlO.  31r 


100  APPLICATIONS   OF   THE   DIPFEEENTIAL   CALCULUS. 

Ex.  15.  Divide  a  into  two  factors  the  sum  of  which  shall  be  a  min- 
imum. Result,  The  factors  are  equal. 

Ex.  16.  The  difference  between  two  numbers  is  a ;   required  that 
the  square  of  the  greater  divided  by  the  less  shall  be  a  minimum. 

Result,  The  greater  =  twice  the  less. 

Ex.  17.  To  find  the  number  of  equal  parts  into  which  a  must  be 
divided,  so  that  their  continued  product  shall  be  a  maximum. 

Sug's, — Tlie  function  is  u=zl-\.     logw  =  a;(loga  —  \ogx).     u    ^=  .^'log«  — 


a; logic.     —  =  log  a  —  log  a;  —  1  ^=  0.     x  =  -.     Arithmetically  the  problem  is 


possible  only  when  —  is  integral. 

Ex.  18.  Eind  a  number  x  such  that  its  ^th  root  shall  be  a  maxi- 
mum. X  =  e. 

Ex.  19.  A  privateer  wishes  to  get  to  sea  unobserved,  but  has  to  pass 
between  two  lights,  A  and  B,  on  opposite  headlands,  the  distance  be- 
tween Avhich  is  a.  The  intensity,  at  a  unit's  distance,  of  A  is  h,  and 
of  B,  c.  At  what  point  must  the  privateer  cross  the  line  joining  the 
lights,  so  as  to  be  as  little  in  the  light  as  possible  ;  it  being  under- 
stood that  the  intensity  of  a  light  at  any  point  equals  its  intensity 
at  a  unit's  distance  divided  by  the  square  of  the  distance  from  the 
light.  . 

6  c 

SuG. — Letting  x  =  the  distance  from  A,  the  function  is  u  =  —  -] — 


x-i   '    (a  — x)2 

J. 


ao 
X  =  — — 


^   3        ,  3 

0       -\-   C 


Ex.  20.  The  intensity  of  illumination  from  a  given  light  varies  as 
the  sine  of  the  angle  under  which  the  light  strikes  the  illuminated 
surface,  divided  by  the  square  of  its  distance  from  the  surface.  Re- 
quired the  height  of  a  light  directly  over  the  centre  of  a  given  circle, 
so  that  it  shall  illuminate  the  circumference  as  much  as  possible. 


Sug's. — Let  /represent  the  illumination  at  P,  which  is  to  be  a 

sin  l_PO 

maximum  ;   PO  =  B  ;  and  LO  =:  .r.     7  =  — ; — .     But 

LP' 

Bin  LPO  =—-=—-.     .•.!=:=—,= -^  ;  whence 

^^        ^^  LP'       ^R.J^x^Y 

dl       (E->-fx2)- —  3a-2(/?2  +  ar2)-  f  i     . 

T,.  =  ^R:^  -I-  a:^).  '  =  «.      im-\-x^-f-Zx'^{R^+x^)l=6 

Ji2  _^  .,.<  _  3.,;:  ^  0,  and  X  =  Bs/\. 


GEOMETRICAL  PROBLEMS.  101 

Ex.  21.  To  find  in  a  line  joining  the  centres  of  two  spheres,  the 
point  from  which  the  greatest  portion  of  spherical  surface  is  visible. 

Sug's. — The  function  is  the  sum  of  the  two 

zones   whose    altitudes  are,   M  D    and  md ; 

hence  we  must  obtain  an  expression  for  the 

areas  of  these  zones      Let  CO  =^  R,  co=:  r, 

Oo  ==  a,  PO  =  X,  and  Po  =  x  =a  —  x.     ^~^   1^  ^  ^ 

From  the  right  angled  triangle   PCO,  R-  =z  Fig.  33. 

]^x 7?2 

DO  X  aJ ;  whence  MD=-R  —  DO=^  — ,  the  altitude  of  the  zone  seen 

X 

fV'rv*        .    rt*3 

on  this  sphere.     In  like  manner  md  = ; — .     Now  the  area  of  a  zone  being  to 

X 

the  area  of  the  surface  of  its  sphere  as  the  altitude  of  the  zone  is  to  the  diameter 

of  the  sphere,  letting  Z  and  z  be  the  zones,  Z  :  4:7tR'^  : :  — — ; :  2E,    .  • .  Z  = 

„    ^Rx  —  R^      A     1  •    Ti  ^     rx'  —r^  a  —  x  —  r 

zTtR .     And  m  like  manner  z  =  znr -, =  Aitr- . 

x  X  a  —  a; 

X  Ji 

Hence,    letting    S   represent    the    function,   we    have,    S  =  27tR'^ -f- 

^      a  —  x  —  r    ,„         ^        R3   ,  r3         dS'       R^  r^  .       , 

27rr-2 ,  S'  =  R^ ^  r^ .     --  =  -; ^-  =  0  ;  whence 

a  —  X  X  a  —  X     dx        x-        {a  —  xy^ 

9  3.  a. 

x=  "         '' —  ■■" 


a.  3. 

)2 


r                     (r^  4-  i?^)2-| 
;  and  the  entire  surface  =  27r  r^  -f-  J^^ • 


r'  +R' 

Since  27tr^  -f-  ^jtR^  is  the  sum  of  the  hemispheres,  S  is  always  less  than  this  sujn 
except  when  a  =  oo. 


GENERAL   SCHOLIUM. 

The  student  should  now  resume  the  study  of  G-eneral  Geometry  at  Chap- 
ter IV. 


OHAPTEE  IE. 

THE  INTEGMAL    CALCULUS. 


SECTION  I. 
Definitions  and  Elementary  Forms. 

157.  The  Integral  Calculus  is  that  branch  of  the  Infinites- 
imal Calculus  which  treats  of  the  methods  of  deducing  the  relations 
between  finite  values  of  variables,  from  given  relations  between  the 
contemporaneous  infinitesimal  elements  of  those  variables.  It  is  the 
inverse  of  the  Differential  Calculus. 

158.  The  Tvttegval  of  a  differential  function  is  another  func- 
tion which  being  differentiated  produces  the  differential. 

150,  IlfltegvatiOTl  is  the  process  of  deducing  the  integral  func- 
tion from  its  differential. 

I(y0,  The  Sign  of  Integration  is  J",  which  is  a  form  derived 
from  the  old,  or  long  s.  It  is  the  initial  of  the  word  simi,  and  came 
into  use  from  the  conception  that  integration  is  a  process  of  summing 
an  infinite  series  of  infinitesimals. 

Ill's. — Suppose  we  have  given  dy  =  -—^ — — .     This  is  a  differential  function, 

and  we  have  given  in  the  equation  the  relation  between  dy  and  dx.  The  Integral 
Calculus  proposes  to  find  the  relation  between  y  and  x  from  such  a  relation  between 
their  differentials  ;   or,  in  other  words,  to  find  the  function  which  being  differen- 

tiated  produces  the  given  differential.     The  function  in  this  case  is  ?/  =  = ,  as 

1  —  iC^ 

will  be  proved  by  differentiating.     The  latter  is  therefore  called  the  integral  of  the 

//      Axidx  ^x^ 

dy==  I 


^xdx 
and  read,   "the   integral  of  dy  equals  the  integral  of     .^    '     ~-_^,  which  equals 


(1  —  X2)2  1—X^ 

Ix 

(1  — "x2)2' 


1—X^ 

The  conception  of  integration  as  a  process  having  for 
its  object  the  summation  of  an  infinite  series  of  infinites- 
imals may  be  illustrated  by  considering  the  area  of  an 
ellipse  as  composed  of  an  infinite  number  of  infinitesi- 
mal segments,  as  represented  in  the  figure.  Let  A  rep- 
resent the  area  of  the  ellipse  ;  whence  cZA  represents  one 
of  the  infinitesimal  segments,  or  elements  of  the  axea. 


DEFINITIONS  AND  ELEMENTABY  FOKMS.  103 

Now  it  is  found  that  dA  =  -{a^  —  x^^)  dx.     By  integration  it  is  found  that  the 

entire  area  is  Ttah,  h  and  6  being  the  semi-axes.  But,  as  the  entire  area  is  the  sum 
of  the  infinitesimal  segments,  the  process  of  integration  may  be  considered  as 
having  for  its  object  the  summing,  or  adding  together  of  all  the  infinitesimals 
which  go  to  make  up  the  entire  area. 

101.  Important  General  Statement. — Strictly  speaking,  there  is 
no  such  thing  as  a  Process  of  Integration.  Whenever  a  differential  is 
proposed  for  integration,  the  first  question  is,  Is  this  a  Knoivn  Form  f 
that  is.  Can  we  see  by  inspection  what  function,  being  differentiated,  pro- 
duces this  ?  If  we  cannot  thus  discern  the  integral  by  a  simple  inspec- 
tion, the  only  question  remaining  is,  Can  we  transform  the  differential 
into  an  equivalent  expi^ession  the  integral  of  luhich  we  can  recognize? 
Thus,  in  any  case,  we  pass  from  the  differential  to  its  integral  by  a 
simple  inspection  ;  and  the  sufficient  reason  always  is,  This  expression 
is  the  integral  of  that,  because,  being  differentiated,  it  produces  it. 


THREE  ELEMEIVTART   PROPOSITIONS, 

102,  JPvop.  1. —  Constant  factors  or  divisors  appear  in  the  integral 
the  same  as  in  the  differential,  and  hence  may  be  written  before  or  after 
the  sign  of  integration  at  2)leasure. 

Dem. — This  is  a  direct  consequence  of  the  fact  that  constant  factors  or  divisors 
appear  in  the  differential  the  same  as  in  the  function  {48). 

lOS*  ^vop,  2, —  To  integr'ate  the  algebraic  sum  of  several  differen- 
tials, integrate  each  term  separately,  and  connect  the  integrals  by  the  same 
signs  as  their  differentials  were  connected. 

Dem. — This  is  a  direct  consequence  of  the  rule  for  differentiating  the  algebraic 
sum  of  several  variables  (51). 

104,  JPvop,  3, — An  indeterminate  constant  must  always  be  added 
td  the  integral  of  a  function. 

Dem.  —Since,  in  difierentiating,  constant  terms  disappear,  in  returning  from  the 
diflferential  to  the  integral  we  have  to  represent  any  possible  constant  terms  by  an 
indeterminate  constant. 

ScH. — The  method  of  disposing  of  this  constant  term,  which  we  usually 
represent  by  C,  will  be  presented  hereafter.*  The  fact  that  there  may  be 
such  a  term  is  all  that  the  student  is  expected  to  see  at  this  point.  To  illus- 
trate, suppose  y  =^  dax^  -\-  12b,  dy  =  ^ax  dx.  Now,  if  the  latter  alone  were 
given,  we  might  see  that  y  =  dax'^  was  its  integral,  since  being  differentiated 
it  would  produce  dy  =  6axdx.  But  so  will  y  =  dax^-\~  any  constant,  as  126, 
or,  as  we  represent  it,  y  =  3«.r2  -J-  G. 

*  Section  VII.,  closing  illustration. 


104  THE  INTEGRAL  CALCULUS. 

TWO   ELEMENTARY  RULES. 

lOS,  R  ULE  1. — Whenever  a  differential  can  be  separated  or  trans- 
formed into  three  factors  ;  viz.,  1st.  Its  constant  factors ;  2nd.  A  vari- 
able factor  affected  with  any  exponent  except  —  1  ;  and  3rd.  A  differen- 
tiat  factor  which  is  the  differential  of  the  27id  factor  without  its  exponent, 
its  integral  is 

The  product  of  the  second  factor  with  its  exponent  increased  by  1, 
INTO  the  1st  or  constant  factor  divided  by  the  new  exponent.* 

Dem.— This  rule  is  evident  from  {162),  and  the  rule  for  differentiating  a  variable 
affected  with  an  exponent  {56..  Thus,  if  y  =  m[/(a;)]«,  dy  =  mn[/(a;)]«-i d.f{x), 
or  mn  X  [/(''''^)]"~^  X  d.f[x)  ;  whence  to  pass  from  the  latter  to  the  former,  we  have 
to  suppress  the  differential  factor,  d.f{x),  increase  the  exponent  n  —  1  by  1  making 
it  n,  and  divide  the  constant  factor  mn  by  this  n. 

In  the  exceptional  case  the  exponent  by  which  we  would  be  required  to  divide 
according  to  the  rule  would  be  1  —  1=0,  whence  the  result  would  be  oo. 

Ex.  1.  Integrate  dy  =  dax^dx. 

Solution,     dy  =  3a  Xx^Xdx;  whence  y  =  jSax^dx  =  — x^  -\-G=  ax^  -f"  G. 

O 

Ex.  2.  Integrate  dy  =  ax^dx. 

/CL 
axHx  =  -x4  _[-  C. 

Ex.  3.  Integrate  dy  =  (a  -\-  3x^y6xdx. 

Solution,  dy  =  1  X{ci-\-  Sx^y  X  Qxdx,  which  corresponds  to  the  requirements 
of  the  rule,  since  d{a  +  3x^)  =  6xdx.     .'.  y  =  J{a-\- Sx^y6xdx  =  i{a  +  3x2j3 -|- C. 

Ex.  4.  Integrate  dy  =  (a  +  ^x^yxdx. 

Solution. — The  differential  of  the  quantity  within  the  parenthesis  being  Qxdx, 
we  write  dy  =  \  {a  -\-  3a;2)3  x  Qxdx,  which  conforms  to  the  requirements  of  the 
rule.     .'.  y  —  f\{a  +  3.r2)36arc?a;  =  -^{a  +  ^x'^Y  +  G. 

Ex.  5.   Integrate  dy  =  a{ax  -\-  hx^y^dx  -\-  '2h{ax  -\-  bx^)^xdx. 

Sug's.  y=fla{ax-{-hx^ydx  +  2b{ax-\-bx-^yxdx']  =  f{{ax-\-lx^)\a-}-2hx)dx'] 
=  /[I  X  (ax  +  hx^f  X  («  +  2hx)dx']  =  i{ax  -}-  bx^^Y  -f-  G. 

100,  RULE  2. — "Whenever  a  differential  can  be  written  in,  or 
transformed  into  a  fraction  whose  numerator  is  the  exact  differen- 
tial OF  its  denominator,  the  integral  is  the  Napierian  logarithm  of 
the  denominator.* 

*  In  giving  such  rules  the  constant  term  of  the  integral  is  not  mentioned,  as  its  addition  is 
always  implied. 


DEFINITIONS   AND   ELEMENT AllY   FOllMS.  105 

Dem. — This  is  a  direct  consequence  of  the  rule  that  the  differential  of  the  Na- 
pierian logarithm  of  a  number  is  the  differential  of  the  number  divided  by  the 
number.     [This  will  be  seen  to  be  the  exceptional  case  under  the  preceding  rule.] 


167.    ELEMENTARY  FORMS. 

1.  V  =    Cx^'dx  =    -^"+^  +  G.     Same  as  Bule  1. 

^        -^  •       n  +  1 

/dx 
—  =  log  X  -\-  G.     Same  as  Bule  2. 

S.  y=   fa'dx  =  -^a^  +  G. 
^  log  a 

3i.  y  =  fe'-dx  =  e'  -\-  G. 


4..  y  =  fcos  X  dx  =  sin  x  -\-  G. 

5.  y  =   r  —  sin  X  dx  =  cos  x  -j-  G. 

6.  V  =   /  >  or  fsec^  x  dx  =  tan  x  -{-  G. 

J  cos^^        -^ 

7.  V  =  / — — >  or  f  —  cosec^^  dx  =  cot^  +  G. 

J        sin2^        -^ 

8.  2/  =   rtan  x  sec  xdx  =  sec  x  -{-  G. 

9.  y  =  f  —  cot X cosec xdx  =  cosec x  -\-  G, 

10.  ?/  =    fsin  X  dx  =  vers  j:  -f  (7. 

11.  y  z=  C  —  cos  X  dx  =  covers  x  -\-  G. 

12.  y  =  I      .     =:  sin~^^  +  G. 

13.  2/  =    / ; =  COS-^^  +  U. 

dx 


14.  1/  =  fzr^-  =  tan-^j7  +  a 

15.  y=    f—  r^-  =  cot-^o;  +  G. 

^        J       1  -{-  x^ 

16.  2/  =  f — 7=^=  =  sec-^^  4-  G. 

J    XV  X'^  ■ 1 

17.  y  =  I —  =  cosec~^a;  +  C, 

J  XV  x-^  1 

18.  ?/  =   /  =  vers~^^  +  G. 

^     Vix  072 

/dx 
.  =  covers-^o;  +  G. 
V  2,r  —  J72 


Converse  of  (60). 

« 

(61), 

(( 

(66). 

« 

(67), 

it 

(69), 

-a    " 

(70), 

<c 

(71), 

tx 

(72), 

a 

(73) 

(t 

(74), 

(« 

(75) 

ee 

(77) 

ct 

(79) 

<c 

CSO) 

tt 

(81) 

« 

(82) 

tt 

(83) 

tt 

(84), 

106  THE  INTEGRAL  CALCULUS. 


168.    SUBORDINATE   CIRCULAR  FORMS, 

r        dx  I    .  _,'bx       ^ 


dx  .     ,x 

—  =  sm    - 

\/a^  —  x^  ^ 


or  =   r       ''"'     -  =  sin-^-  +  a  when  b  =  1. 


=  -  COS  ^ h  u, 


dx  1       _i  &^ 

=  T  COS      ■ 

—  b^x^        ^ 

dx  _,  ^ 


"  ^       ^        v/a2  —  b^x^        ^  « 

or  =  f— -- =  cos-^^  +  C,  when  6=1. 

J         Va;^  —  x^ 


r     dx  1  ,     _^bx 

3.  V  ==    / 7 —  =  -T  tan  '  -  +  0, 

or  =    T-^^  =  -  tan-^-  +  G,  when  6=1. 

J  a;^  ^  x^        a  a 


_  4  cot-  ^  +  c, 


=/-. 


c?.37  1       ,   ,bx 
=  —  cot 

-f  b'^x^        ab 

dx  1      .   ,x 


=  -  cot""^  -  +  0,  when  6  =  1. 


■^  x-^       a  a 

5.  v=    / — 7 =  -  sec [-  Cy 

^       J  Xs/-t)^2oc-2  —  a:^       a  (^ 

or  =    / — ■—- =  -  sec"-'  -  +  (7,  when  6  =  1. 

J  XV  x-i  —  a2        ^  " 

6.  v  =   /  —  — ; —  =  -  cosec h  ^» 

J        xvb-^x'-  ■ —  a'^       ^  ^ 

or  =  f T-''  =  -  cosec-'  -  +  G,  when  6=1. 

/dx  1  1^^  ,  n 
— .^=  =  -r  vers h  <^> 
^2a6a;  — 62^2       o              a 

*  or  =  f—^=Mz=  =  vers-'-  +  C,  when  6  =  1. 

r  dx  1  _,bx 

8    V  =   / =  T  covers h  ^j 

*/         \^2abx  —  b'x^        ^  ^ 

or  =  f ^^         =  covers-'  -,  when  6=1. 

J        Vlax  —  x^  ^ 

Dem.— These  forms  may  be  considered  as  the  converse  of  Ex's.  1,2,  pages  38,  39. 
They  may  also  be  established  by  differentiating  the  result  and  showing  that 

/I    .        hx\ 
its    differential   is    the    given    differential    function.       Thus,    a^^  sm     — )    "^ 


DEFINITIONS  AND  ELEMENTARY  FORMS.  107, 

'\aj  la  1        hdx  dx 


1        \aj  la  _        ^, 

[The  student 


^       r        6^x2        &       ja*  — 62x2        ^  v^o2_52a;2        y/a2_  523.2* 


I         65a;2        6       |a«  — j 


should  verify  all  of  them  in  this  way.  ] 

A  direct  way  of  obtaining  these  integrals,  and  one  with  which  the 
student  should  not  fail  to  become  familiar,  is  the  following  : 

/dx 
— —  ,  we  observe  that  it  has  the  general  form  of  the 

s/a'^  —  &2a;2 

differential  of  an  arc  in  terms  of  its  sine,  which  is  —      -  To  transform  our 

\/l  —  x'-^ 
expression  into  this  form,  we  have  first  to  make  the  first  term  under  the  radical  1. 

This  can  be  readily  done  thus,    /  —  — 

,/   \/a2— &x2 

since  the  constant  divisor  a  appears  in  the  same  form  in  the  integral  as  in  the  dif- 
ferential {102).  Now  to  make  the  quantity  under  the  sign  of  integration  the 
differential  of  an  arc  in  terms  of  its  sine,  the  numerator  ought  to  be  the  differen- 
tial of  the  square  root  of  the  second  term  in  the  denominator,  which  is  the  square 

of  the  sine.     But  d(  —\  =  -dx.    "We,  therefore,  need  to  introduce  -  into  the  nu- 
\  a  /       a  a 

merator.     This  can  be  done  by  putting  -  outside  the  sign  of  integration  as  they 

0 

will  neutralize  each  other  (162).     Hence 

a 


-dx 

— .     The  quantity  now  under  the  sign  of  integration  is  the  exact 


&x  &X 

differentiaLof  sin—'—,  since  it  is  the  differential  of  the  sine,  — ,  divided  by  the 

ti  a  a 

&X  f 

square  root  of  1  — the  square  of  the  sine,  — (75).      Hence,  finally,  as  J  dy=y\ 

,  r         dx  1    .        hx 

we  have  y  =    I  — =  -  sm— 1 [-  C. 

J    \/a2  _  12^2        0  0, 

[The  student  should  produce  all  these  subordinate  integrals  in  this  way,  for  the 
benefit  of  the  exercise.  We  give  the  outline  of  two  more,  which  should  be  ex- 
plained at  length  as  above.] 

h 


y 


/dx        __     /  dx  1     /      dx la    /      a 


at  /  62x2        ab 


ab  i  ^    .   62a;2  ~  ^  ^^     a"  "*" 


108  THE  INTEGRAL  CALCULUS. 


y  — 


/dx /^  dx ^     /*  ^ 


da;  _  dx 


1  /      a /      a 


covers—*  — f-  C. 
a 


169.    LOGARITHMIC   TRIGONOMETRICAL   FORMS. 

y^    dx 
2  cos2  (ix) 
— : — .  r    '"    = 
sm  (ix) 
cos  (|jc) 

/       tdJi{ix)  J      t-d.n{ix)  ^ 

_  r  dx        r       ■,        r      dx  rdiirt  —  x) 

2.  y  =  I  or      sec;rcZa;  =  /  -: ; =  —  /  -: =  [by  (1)J 

^      J   cosx      -^  J   sm(47r  —  x)  J  sin(i;r — x)       »-  ♦'  ^  ^J 

—  log  tan  {i7t  —  ix)  +  ^' 

r  dx  r     .     -,  /"cos  xdx  rd{^\n  a;)  . 

3.  y  =    I or   I  cot  xdx  =    /   — : =    I   — ■ =  log  sm  x  4-  (J. 

I   tanic       -^  /      HVOLX  J      sin  a; 

/^  dx  r  ,  rsiuxdx  /•(i.'cos.r) 

4.  V  =    /  or      tanjcox  =    /  =  —  I  =  — log  cos  a;  = 

/   cotic       ^  J     cos  a;  /      cos  a; 

log =  log  sec  X  A-  €, 

cos  X 

5.  V  =    /  —■ =  I  rr-T  =    I  -■ — rr—  =  \yl  (1)1  log  tan  .r  -I-  C. 

^         J   smxcosrc         /   sm  (^2:c)  /   sm  ^^2^:;)         l  j  v  /j      o  ^ 

ScH — The  above  32  forms  must  be  so  thoroughly  memorized  as  to  be  in- 
stantly recognized.  There  is  no  doing  anything  in  the  integral  calculus 
without  this.  These  forms  are  to  integxation  what  the  multiplication  table 
is  to  arithmetical  operations.  Thus  we  say,  7  goes  into  56  8  times,  because 
8  times  7  =  56.     In  like  manner  we  say  that  cot~'.r  -(-  0  is  the  integral  of 

-,  because  cot~^a;  -{-  C  differeniiated  =  —  :j — -' — -. 


1  +  a;2'  '  •"  1  +  a;2 


Ex.  1.  Integrate  dy  =  'iax^dx. 

SoiiUTioN.-— The  integral  of  dy  is  y,  since  y  differentiated  =  dy.     To  integrate 
Zax^dx,  notice  that  3<2  X  ^^  X  dx,  conforms  to  {105).    ,•.  y  =    fSax^dx  = 

Ex.  2.   Integrate  dy  =  (2a  +  c^hxydx, 

Sug's.     y  =  J {2a  +  3bx)^dx  =  J(8a^  +  36a^bx  -f  BAdb^x^-\-  27b^x^)dx  = 


DEFINITIONS  AND   ELEMENTAKY   FORMS.  109 

fSa^dx  -\- JSGa'^bxdx  +  J^4:ab^afidx  +  j21h^xHx  =  8a^x  +  ISa^bx'  +  ISab'^x^ 
+  Y63a;4  .^  G  (163). 

This  may  also  be  integrated  by  {165).     Thus  y  =  /  oT  X  (2<^  +  3&x)3  X  3&cte 


=/^^(2«+ 


=  j^  (2a  4-  3&a;)4  +  ^  5  which  is  the  same  as  the  preceding. 

xdx 


\/a'^  +  x^ 
Sug's.     y  =  l/^^-  =  f{a^  +  x^)~^xdM  =  /i  X  (a^  +  x^)"^  X  SxcZa;  =s 


2/  = 

=  %hx^ 

+  a 

y  = 

X-' 

+  a 

y 

5 
—  1^^ 

+  a 

y^ 

== 

^mx^ 

+  a 

y  = 

^        1 
Sx^ 

+  a 

Ex.  3.  Integrate  dy  == 

'      xdx 

(a  +  a;2)i  +  a 

Ex.  4.  Integrate  dy  =  bx^dx. 
Ex.  5.  Integrate  6?y  ==  3a7~^c?a7. 

2 

Ex.  6.  Integrate  ^y  =  2x^dx. 

Ex.  7.  Integrate  dy  =  —  5mx~^dx, 

Ex.  8.  Integrate  dy  =  — . 

X* 

x^dx  1 

Ex.  9.  Integrate  dy  = -.  y  =  |.(a»  +  jp3)^  +  G. 

Ex.  10.  Integrate  dy  = dx. 

(Sax-^  —  x^)'^ 

2ax x^  — i  *  _i 

Sug's. jdx  =  —  (Sax^  —  x^)     (2ax  —  x'^)dx  =  _  ^  x  (3aa;2 — x^) 

{3ax-  —  x^)  ^ 

/2ax x^  ^ 
: -dx  =  —  i{dax^  —  x^)''  +  a 
(Sax^—x^)'' 

Ex.  11.  Integrate  dy  =  12bx{4:bx^ — 2cx^)^dx — 9ca?«(46a7« — 2cx^)'^dx. 

Sug's.  126a;(46a;2  —  2cx3)  da;  —  dcx^ibx^  —  2cx^)  dx  =  (4&a2  —  2cx3)* 
(12&a;  —  9cx^)dx.  Now  in  order  that  the  factor  (126a;  —  9cx^)dx  should  be  the  dif- 
ferential of  4bx^  —  2ca;3  we  should  have  8  instead  of  12  and  6  instead  of  9.     Hence 

we  write  f (4&x2  —  2ca;3)  {8bx  —  6GX^)dx.     ,-.  y  =  %{4:hx^  —  2c£c3)*  -f  C. 

170,  ScH. — It  is  not  always  easy  to  determine  just  what  constant  factor 
is  required  in  order  to  make  the  differential  factor  the  differential  of  the 
quantity  within  the  parenthesis ;  nor  can  such  a  factor  always  be  found. 
To  determine  whether  there  is  such  a  factor  or  not ;  and,  if  there  is,  to 
find  it,  we  may  proceed  as  in  the  following  examples. 


110  THE  INTEGBAL  CALCULUS. 

Ex.  12.  Integrate  dy  = '■ jdx. 

(26  +  dax^  —  5a;3)^ 

Stjg's. — jdx  =  (2& -|-  3cwr2  —  s^s)  ^  (2ax — 5x^)dx.    Suppose  A  to 

(26  +  3aa;2  — 5a;3)^ 

/I  -i 

—{2b-{-dax^ — 5x^)    {2aAx — 5Ax^)dx. 

It  is  required  that  A  should  fulfill  the  condition  d{2b  -\-  Zax^  —  6a^)  =  {2dAz—5Ax^)dXy 
or  6ax  —  ISx^  =  2aAx  —  5 Ax-.  Now,  as  this  is  to  be  true  for  all  values  of  x,  we 
have  6a  =  2aA,  or  ^  =  3  ;  and  also  15  =  5A,  or  A  =  d.  Hence  3  is  the  factor 
sought,  and  we  have 

y  ==  J  (2&  +  3aa;2  —  5a;3)  ^ (2ax  —  bx^)dx  =  J  i(26  -f-  3aa;2  —  5x3)  "*  {<oax  —  l^x^)dx  = 
i(26  +  3aa;2  —  5ic3)''  +  G. 

3  3 

Ex.  13.  Integrate  dy  =  x{l  +  a;^  —  2x^Ydx  —  ^x*{l  -\-x^  —  2x^Ydx. 

5.  JL  3. 

Sug's.  dy  =  x{l + x2 — 2x5) ^(ix — 3x4(1 4-x2—  2x5) ^dx  =  (1 + x2 — 2x5) ^ (x— 3x4)dx. 
"We  are  to  seek  a  factor  A,  which  fulfills  the  condition  d(i  -\-  x^  —  2a;5)  = 
(^Ax  —  3Ax'^)dx,  or  performing  the  differentiation,  and  dropping  dx,  2x  —  lOx^  = 
Ax  —  3Ax^  ;  whence  the  first  condition  required  is  A  =  2,  and  the  second  is 
SA  =  10,  or  A  =.  ^£-.  These  conditions  being  incompatible  with  each  other, 
there  is  no  factor  which  meets  the  conditions,  and  the  integration  cannot  be  per- 
formed in  the  manner  now  under  consideration. 


12        3\,  12   ,    1 

X.  y  = 

^  X       x^ 

1  5  3. 


Ex.  14  Integrate  dy  =  (—  —  -)dx.  y  =  —       +      4- G. 


\X^  X*'      'if  X         x^ 


Ex.  15.  Integrate  dy  =  (f aa;^  —  ihx^)dx.        y  =  ax^  —  bx'^  +  C, 

dx  CLX^  1 

Ex.  16.  Integrate  dy  =  dx^dx  -f  — '-pz.  y  =  —- +  x'^  +  C. 

Ex.  17.  Integrate  dy  =  {2x*  —  dx^  +  1)"^(^3  _  ^x)dx. 

y  =  ^{2x^  —  3x^  +  1)^  +  G. 

Ex.  18.  Whicli  of  the  following  can  be  integrated  by  the  method 

used  in  the  preceding  examples  ;  yiz.,  dy  =  {5x^  —  2x)'^{Sx  —  5)dx  ; 

J  J  2x 1 

dy  =  5{5x^  —  2x)'^xdx  —  (5a;2  —  2x)'^dx  ydy  =  — — ^ — — — —dx ; 

dy  =  Tz: ; r<^x  ? 

^      {1  —  X  -\-  x^y 

ScH. — Caution,  The  student  must  not  fail  to  observe  that  it  is  only 
constant  factors  wliich  he  can  introduce  in  the  manner  illustrated  in  the 
preceding  examples.     When  the  differential  factor  is  not  of  the  right  form 


DEFINITIONS  AND  ELEMENTAKY  FOKMS.  |11 

as  to  the  variable^  nothing  can  be  done  with  it  by  this  process.  Were  we 
to  attempt  to  introduce  a  variable  factor  into  the  differential  factor,  its  re- 
ciprocal would  have  to  be  introduced  into  the  1st,  or  constant  factor,  and 
this  would  destroy  the  condition  that  the  first  factor  is  constant.  It  is  always 
well,  if  there  is  the  least  doubt  whether-  the  integral  found  is  correct,  to  differen- 
Hate  it  and  see  if  it  gives  the  proposed  differential. 


17 !•  JPvop, — It  is  sometimes  possible  to  bring  a  differential  to  the 
form  required  in  {lOS)  by  transposing  one  or  more  factors  of  the  va- 
riable from  the  factor  in  the  parenthesis  to  the  differential  factor,  or  vice 
versa. 

Ex.  19.  Integrate  dy  =  -. 

{2bx  +  a;2)3 

Solution. r-    ==    a{2hx  -f  x^)  ^xdse   =   a(2l)ar-i  -f  1)     ^xr-^dx   == 


3. 


(%hx  4-  x-^f 


a 


.a 


X  (.26ar-i  +  1)  ^  X  (—  2&x-5(Za;),  in  which  —  2hx-^dx  is  the  diflferential  of 

(2bx-^  +  1).     .-.  y=  T— ^^^  =  r_ ^  X  (26X-1  + 1)"'  X  (- 2hx-^dx)  = 
J   (2bx4-x^)'     J        ^^ 


b  b^lbx  4-  a;2 

_     ^^    _  ,         ,     ,  adx 

Ex.  20.  Integrate  dy  = 


x\/^bx  +  4c2a;2 
SuG.    — —  =—(3?)a--i-f4r^2)  '"'Sbor^dx,  y  = ^^ ^J- ^  +  (7. 


Ex.  21.  Integrate  dy  =  -?^  {^166).  y  =  log  (1  +  x'^)  +  G. 

ScH. — When  the  integral  is  a  logarithm,  it  is  customary  to  write  the 
constant  also  as  a  logarithm.  Thus,  in  the  above  example  if  we  let  log  c  =  G, 
i.  e.  call  the  constant  term  log  c,  instead  of  C,  we  have  y  =  log  (1  -f-  ^'")  + 
log c  =  log  [c(l  +  x^)],  or  log  (c  +  cx^). 

Ex.  22.  Integrate  dy  = -dxi 

a;2 2x  1  6a;2 12a; 

SoiiUTioN.     Tc—^- — 7-^dx  =  t;  X  K — '■ — 7, —^dx,  vo.  which  the  nmnerator 

2x3  —  &x-  -\-l  6       2x3  —  6x2  -|-  1 

/j)»2 2x 
_ —-^dx  = 
2x-  —  6X--2  +  1 

g-^2 i2x  1  -if 

-dx  =  ^  log(2ic3  —  6a;2  +  1)  +  log  c  —  log  [c(2x3  —  6x2  -f  1)^]. 


1  r  Qx'i 

a  J  2x3  — 


6x2  _}.  J 


112  THE  INTEGRAL  CALCULUS. 

Ex.  23.  Integrate  dy  =  ..^"^'^"l,.        y=  log  [c*(15a;^  +  21)^]- 

Ex.  24.  Integrate  dy  = -. 

1  —  x'^ 

2/=— |log(l— a;^)+logC  =  log[c(l— a:^)    ^]=log 5-^. 

Ex.  25.  Integrate  eZ?/  = r— .  ^:i=log  [c(a  +  2>a?^)]. 

Ex.  26.  Integrate  dy  =        ^     --.  2/  =  log 5:. 

^""'^'^  (8a  — 3a:^)^ 

Ex.  27.  Which  of  the  following  can  be  integrated  by  the  method 

used  in  the  last  6  examples  ;    viz.,  dy  =  — -^ — -dx  ;    c??/  = 

2ir»     ,        ,         a:  —  ^2,         ,            3^'^c?^         ,  2a  —  10a;2 

dx',dy  = -dx  ;    6^2/  =  ^^- ^  ;   ^^Z  =  h" — e:::,^?^? 


1_^3      '     ^        3  — a;3      '      ^        2^='— 5        *"        2aa;  — Sa:^ 
Ex.  28.  Integrate  dy  =^    ^  ' 


nx"^ 


5(3a;  _  a»)4da;       h/^lx*dx  —  108a2a;''da;4-54:a4a;2(^  _  12aSa^a;+a8da;'* 


nx^  n> 


-( Slajcto    —    108a2dx    + —    r—    +    ---  ). 


1 
Ex.  29.  Integrate  dy  =  (h  —  572)3^2,^-^, 


,  =  i63.l_^*-^.HA^-^-,V^+a 


-n,     «^    -r  ,         ,     ,         5(2a  —  ^«)3, 
Ex.  30.  Integrate  dy  =  — -dx. 


y  =  5[-  ^V  ^'  +  6a logx  -  ix']  +  a 


dx 
Ex.  31.  Integrate  dy  =  Slog^o; — . 

Sug's.     2/  =  fs  X  (loga;)2  X  —  =  logs  a;  +  C,  since  —  =  d  logaf. 

dx 
Ex.  32.  Integrate  dy  =  2  logs  -j; — .  2/  ==  i  log"*  ^  -\-  G, 

*  In  all  these  examples  c  represents  the  constant  of  integration. 


DEFINITIONS  AND   ELEMENTARY  FORMS.  113 

doc  7TL 

Ex.  33.  Integrate  dy  ==  mlog"a^ — .  y  = -losf""*'^  x4-  G. 

.  X  n  +  1 

Ex.  34.  Integrate  dy  =  a?""  log  adx. 

Sug's. — In  order  to  make  this  conform  to  {107 f  3),  we  should  have  d{^x)  =.  Idx 
as  a  lactor.  Hence  we  write  ?/  =  \  oP''^  log  adx  =  J^kct-""  log  a  •  2dx  =  ia^-^  -j-  C\ 
[The  pupil  should  differentiate,  verify,  and  so  fully  consider  the  case  as  to  see  the 
reason  for  the  introduction  of  the  constant  factor.  ] 

Ex.  35.  Which  of  the  following  can  be  integrated  by  {107 ^  3)  ; 
dy  =  a"  log  a  2dx,  or  dy  =  da""'  log  a  x  dx? 

Am.,  The  latter,  y  =  ^a'^  +  G. 

Ex.  36.  Integrate  dy  =  e"dx.  2/  ==  ae"  +  G. 

Ex.  37.  Integrate  dy  =  Se^dx. 
Ex.  38.  Integrate  dy  =  hd^'dx. 

Sug's.     y  =  h  fa-'dx  =  — fa^'^  log  a  3dx  =  — a^*  +  C. 

•^  3  log  a*^  3  log  a 

771 

Ex.  39.  Integrate  dy  =  me''''dx.  2/  =  —  e"*  +  <^. 


Ex.  40.  Integrate  cZ?/  =  cos  {2x)dx. 

Sug's. — In  order  to  make  this  conform  to  {107 f  4),  we  should  have  2dx,  i.  e.  the 
differential  of  the  arc  2a;,  instead  of  dx.  Hence  y  =  Jcos  2xdx  =  if  cos  2x  •  2dx 
=  ^  sin  2x  +  a 

Ex.  41.  Which  of  the  following  forms  can  be  integrated  by 
{107 f  4)  ;  dy  ■=  cos^x-ldx,  or  dy  =  cosx^xdx? 

Ans.,  The  latter,  y  =  -^sinojs  +  C. 

Ex.  42.  Integrate  dy  =  sins  ^  cos  xdx. 

Sug's.  y  =  J  sin^  a;  cos  a;da5  =  J  1  X  (sin  x)^  X  cos  xdx  =  4  sin^  a;  +  ^>  accord- 
ing to  {105  and  jer,  4). 

Ex.  43.  Integrate  cZ^/  =  sin  {3x)dx. 

y  =  —  ^  cos  (3^)  +  G,  or  J  vers  (3a:)  +  (7. 

ScH.     i  vers  [Sx]  +  0  =  i  [1  —  cos  (3a;)]  +  0  =  i  —  i  cos  (3a;)  +  G. 

Ex.  44.  Integrate  dy  =  sin2  {2x)  cos  (2^)c?a:. 

2/  =  -|-sin3(2^)  +  G. 

Ex.  45.  Integrate  dy  =  cos^  {Sx)  sin  (3a;)c?j:. 


114:  THE  INTEGRAL  CALCULUS. 

Ex.  46.  Integrate  c^?/  =  sec^x^xdx.  y=^  tan  372+  Q 

Ex.  47.  Integrate  dy  =  5  sec^  x^  •  x^dx,  '    2/  =  f  tan  x^  +  6'. 

Ex.  48.  Integrate  dy  ==  6  sec  (4a;)  tan  {4:x)dx. 

y  =  3.  sec  (4a:)  +  C'. 

Ex.  49.  Integrate  dy  =  2  sin  (a  +  Sx)dx. 

y=^—  |cos(a  +  3d7)  +  C. 

Ex.  50.  Integrate  dy  =  ^  cosec^  v2x  ■  x  '^dx. 

Sug's.      «    =     ff  cosec^  \/2a;  •  ic    dx    =   — =  fcoseca  v^  •  ^>/2  •  a;    dr    = 
^  *^  v/2 

— 2_  cot  v^2x  +  a 

v/2 

Ex.  51.  Integrate  dy  =  2  cosec  (na;)  •  cot  {nx)dx. 

y  = cosec  (nx)  +  G. 

Ex.  52.  Integrate  dy  ==  e"'" "  cos  ajtZa;.  y  =  e"""  +  C'. 

Ex.  53.  Integrate  dy  =  —  e"'""' sin  xdx.  y  =  6"'""  +  C. 

dx 
Ex.  54.  Integrate  dy  =  — 7TT~T'  2/  ==  2  tan  {^x)  +  G. 

cos^  (-^a^j 

xdx 
Ex.  55.  Integrate  c^v  = .      -^  on-  1/  ==  i cot  (3a;2)  -f  a 

°  sm"  (da72) 

Ex.  56.  Integrate  dy  =  sin  {ax)dx. 

V  =  -  versin  (ax)  4-  G,  or cos  (ax)  -\-  C. 

^        a  a 

Ex.  57.  Integrate  dy  =  —  cos  {^x^)xdx. 

y  ==  covers  {^x^)  +  G,  ov  —  sin  {^x^)  +  C". 

ScH.— In  the  last,  C"  =  C+  1.     In  the  56th  Ex.,  G'  =  C  +  -, 

Or 


2dx 
Ex.  58.  Integrate  dy  = 


V/I   —   4:Xi 

Sug's.  —  The  form  of  the  denominator  suggests  at  once  that  this  may  be  the  dif- 
ferential of  some  arc  in  terms  of  its  sine.  Observing  that  the  numerator  is  the 
differential  of    the   square   root   of  4ic'',  we   are   enabled   to   conclude   that  y  = 

=  sm-i  2x  H-  a 


/; 


\/l  —  4a;2 

xdx 
Ex.  59.  Integrate  dy  =  .  y  ===  ^sin""^  {x-)  -f  0. 

V  1  —  a:^ 


Ex.  60.  Integrate  dy 


DEFINITIONS  AND   ELEMENTARY  FOEMS.  115 

dx 


^2  —  9^2 


—  -dx 

^     ,  dx  dx  \/2  >/2  /•      dx 

SUGS. 


v/2  — y.r^      v/2v/l— |a;^        ^      v/2>/l  — fx"^  ^   >/2— 9x-J 


^    >/2 


/    — -dx 
.1      /      v/2 


1   .         3cc    ,    ^ 


Qcdx 

Sx.  61.  Integrate  dy  = 


\/^  —  5x4 


„     ,  /•    —  xcZx  /*       —  .r  Ja;  /*        — 2\''^xdx 

ScTGs.    y  =   I  —  =   /  — = — -  =   /  — = — - —  =5 

J    v/2  —  5.r^        J    v/2  yi  —  fx^        ^   2v/f  v/'2  v/1  —  f^t^ 

1     /-—  'l^l.xdx        1  ,  r/5\ A    n  ,   ^ 

—  /  — =    — ^cos-i     (  -  )  a;H  +  C. 

/5^    v/1  — fx^        2>/5  L\^/      J 


2v/5 

Ex.  62.  Integrate  d/y  = 

Ex.  63.  Integrate  dy  = 

Ex.  64.  Integrate  c??/  = 

Ex.  65.  Integrate  dy  = 

Ex.  66.  Integrate  dy  = 


'6dx 

4  +  9^2* 

,  ,       ,  Zx       „ 
2/  — ^tan-^y  +  C/. 

j;^c?x 

2/  — ^sin-X2^/2-)  _|_  c. 

>/2  —  4^3 

xcfj^ 

2/  —  ^    sm  '  —  4-  a 
26              a 

v/qj2    1)2^4 

x"dx 
1  4-  x^' 

2/    ..  Jtan-Va^s  4-  (7. 

^xr'^dx 

Ni  2a-^  — 6^^ 


Sug's. — As  far  as  the  variable  is  concerned  this  conforms  to  the  diflferential  of 
au  arc  in  terms  of  its  versed  sine.     Thus,  the  numerator  =  d\]  6a;*,  as  far  as  the 

variable  is  concerned  ;  and  x^  =  (x^  )2,  which  is  the  relation  between  the  functions 

of  the  variable  in  the  denominator  of  the  form  referred  to.     Hence,  if  we  can  adjust 

the  constant  factors  to  this  form,  the  integral  will  be  apparent.     To  effect  the  latter, 

we  proceed  as  follows  : 

-I  --2  -^ 

S.'T    dx  .-  8.r  ^dx  ,   /^  2x  ^dx 

=  vo  — -     -  -  --  ■■■■■=: 4v/6 


\  2x '  -  6a;'  Nj  2  •  6x '  —  (6x' y-  N/  2  •  6a; '  —  {Gx'y 

in  which  6x    being  regarded  as  the  variable,  the  expression  has  the  desired  form. 
.-.  2/  =  4v/6  vers-i  (6a;*)  4.  C,  - 


116  THE  INTEGRAL  CALCULUS. 

Ex.  67.  Integrate  ay  =  — y=z=z .  y  =  -7=  sec~^  — r-  4-  G. 

XV  3x^  —  5  V  5  5t 

Ex.  68.  Integrate  dy  = 


\/l4:a;2  —  3 

y==^^osec-^[{^fx]  +  a 


MISCELLANEOUS  EXERCISES  UPON  THE  ELEMENTARY  FORMS. 

[NoTK. — The  following  exercises  are  given  without  the  integrals,  as  it  is  of  first  importance  in 
the  Integral  Calculus,  that  the  pupil  be  able  to  discover  in  the  differential  the  probable  form  of 
the  integral.] 

■^     ^    n-  ,         ,7          (1  —  smx)dx 
Ex.  1.  Integrate  ay  = ; . 

°  ^  X  -\-  cos  X 

Ex.  2.  Integrate  dy  =  (2x^  +  x~'^)dx. 

-r-,     «    -r  ,         ,     -,            ^dx        -       -  x^dx 

.  Ex.  3.  Integrate  dy  =  -— ;  also  dy  •= 


Ex.  4.  Integrate  dy  = 


1  +  X*'  ^  ^    1  4-  ar* 

xdx 
\/l  —  x^ 


qqz ^x  A-  3 

Ex.  5.  Integrate  dy  =  -^—^^^^^dx. 

-r^  «        -r      .  .  •,  ^dX  ,  ,  xdx 

Ex.  6.  Integrate  dy  =  ^   ,    ^     ;   also  dy  =  ^^   ,    .^    . 

^  2  +  5^72  ^       2  +  6ar« 

Ex.  7.  Integrate  c?^/  =  (1  +  cos  J7)<^a:. 

(a  4-  vxydx 
Ex.  8.  Integrate  dy  = 7= . 

V  X 

(2a2  +  4j72)dci7 
I  Ex.  9.  Integrate  dy  = — . 

V  a2  +  x^ 

Mx 
Ex.  10.  Integrate  dy  = 


Ex.  11.  Integrate  (Zt/  =  cos^  x  sin  a;<ia7. 

Ex.  12.  Integrate  dy  =  ion^  x  BeC^  x  dXy  or  (tansa?  +  tan*ar)<Zar. 

&  3 

Ex.  13.  Integrate  dy  =  {a -^  +  cx'^)dx. 

Ex.  14.  Integrate  cZy  =  (1  4-  ^)(1  —  x^)xdx. 


RATIONAL  FBACTIONB^  117 

,Ex  15.  Integrate  dy  =  ^.  -  f^y.^-/  i  :^J  Jl^ 

Ex.  16.  Integrate  dy  =  j~j-;y,. 

{a  —  x)dx 


Ex.  17.  Integrate  dy 


{"lax  —  ^72)^ 
^x^dx 


Ex.  18.  Integrate  dy  =  ^^^^  ^- 

Ex.  19.  Integrate  dy  =  ^ — ; ; . 

^  ^        1  +  a:  +  ^2 

da;  ^dx  4dx  4di»? 

Solution. 


14_a;4-a;2        4  +  4x4-4x2        3  + 1  +  4a;  +  4ii;2        3  _|- (i  _{_ 2a;)2 
i-! ,  which  is  the  form  for  the  differential  of  an  arc  in  terms  of  the  tan- 


•  1  4-2x^2 

1  -|-  2a;      Mx  /*       dx 

2      2da; 


gent,  except  that  the  numerator  should  be  d.    -  ~^r~  =  ~~"r*    • '  •  ^  =  /  1".       1     2 

gS  32  J      -j-x-t-x 


3' 


?7i  -4-  71^ 
Ex.  20.  Inteerate  dy  = ^o?. 

^  ^         a2  _|_  ^2 


-♦-♦-^- 


SUCTION  IL 
Rational   Fractions. 

SEPARATION  INTO  PARTS  BY  INDETERMINATE  COEFFICIENTS. 

[Note. — The  body  of  what  is  called  the  Integral  Calculus  is  made  up  of  Special  Expedients  by 
means  of  which  differentials  of  various  forms  can  be  reduced  to  equivalent  known  or  elementary 
forms.  A  few  of  the  more  important  and  fundamental  of  these  processes  are  given  in  this  and 
the  two  succeeding  sections.] 

172,  Def. — A  national  Fractiofi^  as  the  term  is  used 
here,  is  a  fraction  in  which  the  variable  is  affected  with  none  but  pos- 
itive, integral  exponents.     The  general  form  is,  therefore, 

ax""  4-  hx"^-^  4-  cx"'-^ -  -lx-\-k 

mx"  -f  7ia?"~^  +  p^"~^ rx  -{-  s 


118  THE  INTEGBAL  CALCULUS. 

173,  Prop*  !• — ^  the  highest  exponent  of  the  variable  in  the 
numerator  of  a  rational  fraction  is  greater  than  that  in  the  denominator, 
the  fraction  can  always  be  converted  into  an  equivalent  expression  con- 
sisting of  a  series  of  monomial  terms  with  or  without  a  rational  fraction 
{as  the  case  may  be),  in  which  fraction,  when  it  occurs,  the  highest  ex- 
ponent of  the  variable  in  the  numerator  shall  be  at  least  1  less  than  the 
highest  exponent  in  the  denominator. 

Ill's,     -— i— ■ —  =:  x  —  a  -] 1— ^ .     Again  = 

x2  —  1  '  a;2  —  1  *^        x  —  a 


174:*  l^VOp*  2* — Whenever  the  denominator  of  a  rational  fraction, 

as  — — — ,  whose  numerator  is  of  lower  dimensions  than  its  denominator, 

cp{x) 

is  real  and  resolvable  into  n  keai.  and  unequal  factors  of  the  first  degree, 
the  fraction  can  be  decomposed  into  n  partial  fractions  of  the  form 

Adx  Bdx  Cdx  Ndx 

j 1 

X -h  a       x  +  b       x  +  c  X -1- n 

and  these  fractions  integrated  separately. 

Dem. — Assume    ^^-^  =  — ■ —   A r—r  A ; — — ; — .     Bnngmg 

(p{x)        x-\-ax-\-hx-\-c  X  -\-  n 

the  second  member  to  a  common  denominator,  each  numerator  will  be  multiplied 
by  n  —  1  factors  of  the  form  x  -\-  m,  and  hence  will  be  of  the  (n  —  l)th  degree. 
Collecting  the  coefficients  of  x'^~^,  a;"""^,  etc.  in  the  numerator  of  the  second  mem- 
ber, it  will  take  the  form  3fa;»-i  -f  ^x''-^  -\ Px  -f  Q,x%  in  which  M,  N, P, 

and  Q  are  functions  of  A,  B,  C,  etc. 

Now  fyX)  ^=-  Mx"—^  -\-  iVic"— 2  _j Px  -\-  Qajo,  since  the  denominators  of  both 

members  are  equal.  Then  as /(x)  is  not  above  the  (n  —  l)th  degree,  it  can  be 
treated  as  a  complete  polynomial  of  this  degree  ;  and  the  coefficients  of  the  like 
powers  of  x  being  equated  by  the  principle  of  indeterminate  coefficients,  there  will 
result  n  simple  equations  between  A,B,  C,  etc.,  from  which  these  coefficients  can  be 
determined.  The  values  of  A,  B,  C,  etc.,  thus  determined,  being  substituted  in 
the  assumed  series  and  the  factor  dx  introduced,  the  decomposition  is  effected, 

/Adx 
■      ' 
X  ~j~  ^ 

=  ^log  (x  -|-  «).  6^0. 


17 S»  ^VOp»  3*— Whenever  the  denominator  of  a  rational  fraction, 

as  --— — ,  whose  numerator  is  of  lower  dimensions  than  its  denominator, 

q)(x) 

is  real  and  resolvable  into  n  real  and  equal  factors  of  the  first  degree, 
the  fraction  can  he  decomposed  into  xl  partial  fractions  of  the  form 


RATIONAL  FRACTIONS,  119 

Adx  Bdx  Cdx  Ndx 


(x  +  a)"  '    (X  +  a)"-^       (X  +  a)"-2  x  +  a' 

and  these  fractions  integi^ated  separately. 

Dem.— Assume  "^-^-^  = ; — •-  A ,  4 ■ .     To 

cp'^x)       {X  -j-  a)"   '    (X  -\-  a/*-i  ^  (x  +  a/'-^  x -\- a 

bring  the  terms  of  the  second  member  to  a  common  denominator,  we  have  to 
multiply  Bhy  X  -{-  a,  Chy  (x  -\-  a)'^ and  N  by  (jc  -\-  a)»— i;  hence  the  nu- 
merators, when  added  together,  will  make  a  polynomial  of  the  {n  —  l)th  degree. 
Equating  this  numerator  with  f{x),  the  coefficients  of  the  corresponding  powers 
of  X  being  placed  equal  to  each  other  will,  as  in  the  preceding  demonstration,  give 

rise  to  n  simple  equations  between  A,  B,  C, H,  from  which  the  latter  can 

be  determined. 

Having  separated  the  fraction  into  the  partial  fractions  as  proposed,  it  remains 
to  be  shown  that  these  can  be  integrated.  A«  the  numerators  are  constant,  and 
the  general  form  of  the  denominator  is  {x  -j-  a)»,  we  are  to  show  that  the  form 

r^ can  always  be  integrated.     If  n  =  1,    /  — ; —  =  log  (x  4-  a).     If  n  is 

{x  -1-  «)»  "^  ®  '  J  x  -\-  a  &  V     I      ; 

/ctx               c                                      1 
— -- — -  ==  J  (a;  4-  a)-"dx  = {x  -\-  a)— "+S  or 
( ir  ~p*  Ctj                                                             10  ^~*  1 

; .     Hence  the  integration  can  always  be  effected. 

(w  —  l){x  -4-  a;"-^  ^  •' 

ScH. — The  last  Iavo  propositions  are  equally  true  whether  the  factors  of 
q){x)  are  real  or  imaginary ;  but  as,  in  the  latter  case  the  integration  by 
those  methods  would  give  logarithms  of  imaginary  numbers,  the  method 
given  in  the  next  proposition  is  preferable.  We  will  still  farther  premise 
that  as  <p{x)  is  real,  if  it  contains  imaginary  factors,  they  must  enter  by  pairs 

of  the  form  x  ±:  a  -{-  hV — 1  and  .r  zfc  «  —  h^/ — 1,*  for  only  thus  can  a  real 
product  arise  from  imaginary  factors.  If  therefore,  there  are  imaginary 
factors  in  cp[x)y  we  shall  have 

cp{x)  =ip{x)  •  {{x±:a-{-b^^)  X  {x±:a  —  hv'~i)y^i>{x)  •  { (.r  zb a)^  +  _H "» 
in  which  il){x)  represents  the  product  of  the  real  factors. 


176.  I*VOp,  d, —  Whenever  the  denominator  of  a  rational  fraction, 

f(x)dx 
as       ,  —,  whose  numerator  is  of  lower  dimensions  than  its  denominator, 

is  real  and  resolvable  into  n  real  and  equal  quadkatic  factors,  the  fraction 
can  be  decomposed  into  n  partial  fractions  of  the  form 

(Ax+B)dx  (Cx  +  D)dx_  ,       (Ex  +  F)dx  (Mx+N)dx ' 

r(i:±a)2-fb2]«"^[(x±a)2H-b'^]"-^"^  [(xdba)2+b2]"-^  '  "  "  (xrta)2-f  b^' 
and  these  fractions  integrated  separately. 

Dem.  — [The  first  part  of  the  demonstration,  showing  that  the  fraction  can  be 


*  Called  conjugate  imaginary  faetors. 


120  THE  INTEGRAL  CALCULUS. 

separated  into  this  form,  is  identical  with  that  of  the  last  proposition,  and  the 
student  can  supply  it.] 

Having  separated  the  fraction  into  partial  fractions  as  proposed,  it  remains  to 
be  shown  that  these  partial  fractions  can  be  integrated.     The  general  form  is 

— ~ ; — ,  in  which  n  is  an  integer.     To  reduce  this  to  known  forms  put 

i{x  ±z  ay^  +  &2]«'  s 

X  dz  a  =  z,  whence  x  =  z  =p  a,  dx  =  dz,  and  (x  ±  a)^  =  z"^.     Substituting  these 

values,  we  have 

Aa)dz 


r  {Ax  +  B)dx     _    r^Az  ^  Aa  -\-  B)dz  _    r     Azdz r{B  ^  Aa)di 

J  A{z-^  +  h-^)-«zdz  +    /  ■^f:y^,,  ill  which  A'=  B  =F  Aa. 

By  {165)  we  have  f  A{z^  +  l^)--zdz  =  -  ^^^  _ -^^^^,  _^  j^,y,^,- 
*In  a  subsequent    article   {192,  formula   5|»   it  will  be   shown    that    the 
1 may  be  made  to  depend  upon    /  f-7 -,  which  in  turn  may  be 

(22   _|_    62)n  ^  ^  ^         J     (22   -^    b-^)n-l 

r      A^dz        ^,        .    ^,         ...        ..-./*    A„+idz 
made  to  depend  upon   /      ,    ,    ^,^„_2>  thus,  m  the  end  giving  either   /   ^^,  _^  ^,^„_„ 

177.  ScH. — In  case  the  factors  of  the  denominator  are  not  readily  seen, 
put  it  equal  to  0,  and  solve  the  equation  for  the  variable.  According  to  the 
theory  of  the  composition  of  equations,  as  developed  in  Higher  Algebra, 
the  variable  minus  each  of  the  several  roots  in  turn  will  be  the  factors. 

Ex.  1.  Integrate  ay  == —r -^p: — — -r.. 

Solution.— Putting  x^  +  Qx'^  -f  llic  +  6  =  0,  we  find  cc  =  —  1,  —  2,  and  —  3.+ 

.-.  ic3  -f  6a;2  -f  llo;  -f  6  =  (.r  +  l){x  +  2){x  +  3),  and  we  assume 

''  +  '  ^   +^,+    "   - 


^A  _|_  o.'ra  -]-llx+6~ic+l~a;  +  2        .t  +  3 

A(x  +  2)(cc  +  3)  J3(a'.  +  l)(x.  +  3)  C(a;  +  l)(x  -f-  2^ ^ 

(a;  4-  l)(a;  4-  2)ia;  +  3;  "^  (a;  -f  l)(x  +  2)(x  -j-  3)  "^  (x  +  l)(a;  +  2)(x'+  3) 
^x^  4-  hAx  4-  6^  4-  Bx^  +  4.B.r  4-  3^  +  Cl-r^  +  BC-g  +  26' 
£C^  -}-  6.^2  4-  llx  4-  6 

Whence  x2  +  1  =  (A  +  5  4-  (7).x2  4-  (5^  +  45  4"  3C)x  -|-  6^  +  35  4-  2a 

These  members  being  identical, 
A^B^Q=\    U);     5^1+45  4-3(7=0    (2);     and  6^  4- 35 +  2(7=  1    (3). 

From  (1),  (2),  and  (3)  we  find  ^  =  1,  5  =  —  5,  and  C'=  5. 

Hence  we  have 


*  This  reduction  might  be  exhibited  here,  but  as  the  formula  referred  to  is  better  for  practical 
puriio.es,  it  is  thought  best  to  give  the  process  but  once, 
t  Complete  School  Algebba,  Part  II.,  {ill). 


RATIONAL  FRACTIONS.  121 

{  cfa  +  l)(x  +  3)5 ) 
51og(a;  +  2)  +51og(a;  +  3)  4-logc  =  log]        ^  ^  2^         \' 

adx 
Ex.  2.  Integrate  a?/ 


r    adx     _  1   r  dx  1   r  <f j;    ,         Ic^i^x  —  a 

=  J  x'^  —  a^'~^Jx  —  a~2jx  +  a~  ^^N     a;  +  a  ' 


(3^  —  b)dx 
Ex.  3.  Integrate  ^y  =  ^..g^^s' 

y  =  |log(a;  — 4)-ilog(a;_2)+logc=log4£i^I=:il-[. 

t.(^_2)'5^) 

Ex.  4.  Integrate  efa/  =  ^,  ^  ^^' 

Sug's     — ^^ —  =  —  -1 7  gives  a  =  -45,  and  d  s=z  A-\-B\  whence  -4  =  r-, 

a;2  4-  6x       ic*       jc  4-  0  o 


andJB==-^.   2/=log{c(^^)p 


(2  +  3a?  —  4.x^)dx 
Ex.  5.  Integrate  (^j/  = 1^  _  ^3 • 


2/  =  ilog^  +  flog  (2  +  ^)  +  log  (2  —  a;)  +  a 


5 


Ex.  6.  Integrate  ..  =  g±^^.  y^loA^^ZZ^ 

(5j;  +  V)dx 
Ex.  7.  Integrate  dy  =^  ^,  _^  ^  _  2' 


o 3^2 

Ex.  8.  Integrate  cZy  =  j^      2)»^^* 

2-3x2  ^       ,       ^       ,       (7__£j-5(^+2)+£(H:2)2. 

SuG's.-Assume  -^^-^p  =  ^^::f:2j3  +  (^+2p  +  x+2  "  (x  +  2)3 

whence  2 -3a;2  =  ^  +  jBa:  +  2J5  +  ac'^  + 40^  +  46'.     .-.  ^+2B+40=2  (1); 
^_|.4C'=  0  (2) ;  (7=  —3  (3)  ;  from  which  JB  =  12,  ^  =  —10,  and  G=  —3. 

-H---3iog(a;  +  2)  +  a 
a;  -f-  2 


*  This  factor  is  equivalent  to  a;  -f  0,  so  that  x2  4-  fc*  «=  (»  -}-  0)(x  +  6;. 


122  Tm  INTEGRAL  CALCULUS. 

Ex.  9.  Integrate  dy  =  - — '■ — -^dx. 

2/ =  - -^T.  +  3  log  (a?  —  3 )  +  a 
oc  —  o 

Ex.  10.  Integrate  dy  =  -. —■, — ; — r. 

,^  ^  ^        {x  —  aY{x  4-  a) 

SuG. — AVhen  some  equal  and  some  unequal  factors  occur  in  the  denominator, 

X- 

the  assumed  forms  must  be  combined.      Thus •— ; must  be  put 

(X  —  a)2(x  +  ^) 

==-_i       +_^4.-A_.     ^  =  ia,  J5==|,  andC=i.    y==-—^—-^ 

(x  —  a)-   '    x  —  a      £c  +  a  K^  —  a) 

I  log  (a:  —  a)  +  ilog  {p;  -}-  a)  +  a 


1 


^2  ^^  _}_   3 

Ex.  11.  Integrate  dy  =  __^_^^<;x. 

SuG.     a;3  _  6x2  4.  9a;  =  a;(a;2  —  6a;  +  9)  =  x{x  —  3)2. 

2/ =  i  log  JP -h  ^  log  (a?  —  3)2  +  ^  log  c  =  log  [<^(a;  —  3)2^ . 

Ex.  12.  Integrate  dy  =  (^  _  ^^^(^  ^  g),- 

Sro.-Assume  ^_^2__-  =  _£^-  +  ^^  +  (si)5  +  iTS  '  '''''°" 
4  =  A.  £  =  -  tIs.  C  =  s^s,  and  D  =  tIt- 


Ex.  13.  Integrate  dy  =  —. ,  ^.    fe 

SxjG.— The    simple    binomial    factors   of  the    denominator  being    imaginary, 

,  .  _^  y_  2;2(a;  _  v/-  2)2,  ^^e  assume  ^^^^^qj^F  =  l^^T^^^  +  ^Hfll  '  ^^'^^^^^ 
^  =  —  1,  J3  =  —  1,  C=  1,  and  2>  =  0. 

'     da;  ==    / dx  4-    /  — =  /    — x:x--\-  2)-2da; 

_  /  £ —  =  X  log  (x2  +  2)  -I /  — — — rr-.     The  last  term  can 

J   (x-^  +  2)2       ^     ^'^     ^    ''^2(a;2  +  2)      J   (0:24-2)2 

be  integrated  by  formula  ^  {192,  Ex.  10). 

dx  - 

Ex.  14.  Integrate  dy  = —7; r. 

Sug's  —The  mmple  factors  ar#  a?  +  v/^2,  fl?  —  >/  — 2,>n.d  »  —  1 ;  and  the 


RATIONAL  FBACnONS.  123 

_^x  -\-JB       0 
form  of  the  partial  fractions  is  -'    '     . .     A  =  —  h  B  =  —  i.  and  (7=  ^. 

x--f2     x  — 1 

/dx  ,  /T  xdx        ,  /•   diT      .    ,  /*  dx 


1.      ,  ^    ,  n     '        r  dx          1  « 

—  tan— ^ h  (7 ;  since  /     „  ,   ^  = tan— i  — . 


Ex.  15.  Integrate  t^2/  = 


x^  +  X^  -\-  X  +  1* 


Ex.  16.  Integrate  c?v  = ^• 

SuG  s.  —Assume ; =  — r—  -\ -\ — -  ;  whence  A  =  —  *• 

\x-f-l/  6  ^2 

X^  -\-  X 

Ex.  17.  Integrate  dy  = ; -dx. 

Ex.  18.  Intesrrate  dy  = -dx. 

^  ^        x^  +  x^  +  X  -}-  1 

-p     ia    T  .        .    ^  9^2  +  9^-128     , 

Ex.  19.  Integrate  dy  =  —z --., — ^r ax. 

&  ^       x^  —  5x^  i-  dx  -\- d 

x^  —  1 
Ex.  20.  Integrate  dy  =  -^ jdx. 

!/  =  f-'  +  I  log  (07  +  2)  +  I  log  (^  —  2)  +  a 

17 S,  ScH.— It  will  be  observed  that  the  foregoing  processes  of  separa- 
ting rational  fractions  into  partial  fractions  make  their  integration  depend 
on  one  or  more  of  the  following  forms  : 

/^  r  dx       r   dx       r  xdx       r    xdx         r     dx 

-^  ^     ^'  ./   X  ±:a    J  x^  -\-  a2'  J  x^  +  a^'  J  (.x-2  4.  a^y^'  J  [x^  +  a^)^'  ^ 

All  of  these  forms  except  the  last  are  integrable  by  the  elementary  pro- 
cesses.    The  integration  of  the  last  is  effected  by  formula  i|>  {192). 


4-'^-^   A^ 


-    CT$ 


124:  THE  INTEGBAL  CALCULUS. 

V      SECTION  IIL 
Eationalization. 

170*  When  polynomial  radicals  occur  in  a  difterential  wliicli  we 
desire  to  integrate,  it  is  sometimes  possible  and  expedient  to  rational- 
ize the  expression  by  the  substitution  of  a  new  yariable  which  is  soiue 
definite  function  of  the  variable  in  the  given  differential.  A  few  of 
the  more  important  cases  are  given  in  this  section. 

BINOMIAL    DIFFERENTIALS. 

180.  I^TOp*  1. — Every  binomial  differential  can  he  reduced  to  the 
form  x™(a  +  bx°)Pdx,  in  which  m  and  n  are  integral,  and  n  positive. 

Dem. — 1st.  If  X  occurs  in  both  terms  of  the  binomial,  and  the  form  is 
x^{(ix^  -\-  hx*)Pdx,  we  can  remove  from  the  parenthesis  the  factor  x",  or  a',  which 
has  the  less  exponent.     Thus  suppose  s <^t,  we  can  write  x^^ax'  -j-  hx^) p dx  = 

(x^\p 
a  -\-h—  ]  dx  =  x^+p^{a  -\-  bx^-^)pdx.     In  this  form  t  —  s  is  positive,  since 

t^  s,  but  it  may  be  fractional,     r  -{-  ps  may  be  either  positive  or  negative,  integral 

f  6 

or  fractional.      Now  let  r  -\- ps  =  ±  f,   and  t  —  s  =  -[-  -  ;    whence  we    have 

/I  / 

X   \a+bx    fydx. 

±-  +- 

2nd.  In  the  latter  form  put  x  =  z^f  ]  whence  x   ^  =  z^*^,  x  ^  =  z+^\  and  dx  = 

±-  +- 

hfz^-f—^dz.        Substituting    these    values,     we     have    x    '\a    -j-    ^^  ^^dx    = 

z±c/(a  4-  hz+^'')Phf  z^f-'^dz  =  hf  z±'^f+^f-'^{a  -J-  hz+^^)Pdz,  in  which  the     exponents 

of  z  are  integral,  since  c,  e,  /,  and  h  are  integers,  and  eh  is  positive.     Therefore 

putting  ±  cf-{-hf —  1  =  m,  and  eh  =  n,  we  have  hfz^{a-\-  bz'')pdz.     q.  e.  d. 


181.  JProp.  2. — A  binomial  differential  oftheform'K^(a.-{-hx''ydx 

{any  or  all  the  exponents  being  fractions)  may  be  rendered  rational  by 

.                    ,       m  +  1  .    .   ,        , 
putting  a  +  bx   =  z'^,  when is  integral. 

Dem. — Putting     a  -{-  hx^  =  z^^  (1) 

p 

we  have                     («  +  &  iK")^  =  ^,  (2) 

Differentiating  (1),  nh  x'^—^dx  =  qz^—^dz.  (3) 

m — n+1 

Also  from  (1),  a;'»^+i  =  (J    T     J    "    *  ^^^ 

m — »+l 

Multiplying  (2),  (3),  and  (4),  n&x'"(a  -f  &x")?dx  =  gz^+g-/^  Tj     "     ^2» 

rr/-l-l 

?  n  /z''  —  fl\~t        ' 

or  .r"'(a  +  hx")\1x  —  -^z''  •^''-     — ■ —  ) 

nh  \      b      / 


d.z. 


RATIONALIZATION.  125 

m  -j—  1 
Now  by  hjrpothesis  p  -{-q  —  1  is  integral ;  hence,  if ' —  is  integral,  the  expression 

is  rational,     q.  e.  d. 


182,  JPvop*  3, — A  binomial  differential  of  the  form  x'"(a4-bx")^dx 
(any  or  all  the  exponents  being  /inactions)  may  be  rendered  rational  6?) 

11^  +  1       P  .    .   . 
putting  a  +  bx"  =  z'^x",  when [-  ~  is  integral. 

(1) 
(2) 


Dem.  — Putting 

a  -\-hx'^  =  z?x", 

e  have 

a 

T*"  — 

22  —  6* 

1 

a" 

1 

f  .                        n^(''1 

-h)  ». 

(z?  _  ly 


(3) 


and  jc"  =  a  "(z?  —  6)    «.  (4) 

Multipljdng  (2)  by  Z)  and  adding  a,  we  have 

IT               ah       ,               az^  _. 

a  4-  occ"  = 4-  a  = -,  (5) 

whence  (a  +  6a;")*  =  a9(z3  —  6)    «zp.  (6) 

9  -  ---1 

Differentiating  (3),  do;  =  —  -a^zi  —  6)    «    zs-'dz.  (7) 

Multiplying  together  (4),  (6),  and  (7)  and  putting  A  for  the  constant  factor,  there 
results 

a;"i(a  _{_  'bx^'Ydx  =  ^(z«  —  6)    ^    "       «      ^z?  +  «—  'dz. 

772      1  ■   1  Ti 

Now  by  hypothesis  p  -\-  q  —  1  is  integral ;    hence,  if  — — f-  -  is  integral,  the 

expression  is  rational,     q.  e.  d. 

183,  ScH. — ^Although  the  rationalization  can  always  be  effected  as  stated 
in  the  last  two  propositions,  it  does  not  always  facilitate  the  integration. 

When  in  the  former  case  — — 1  is  a  positive  integer  or  0,  or  in  the 

Wi     I      J.  T) 

latter 1-  -  +  1  is  a  negative  integer  or  0,  the  binomial  ^  —  b  will 

71  q 

have  a  positive  integral  exponent  and  can  be  expanded  into  a  series  of  a 

finite  number  of  terms,  or  a  0  exponent  and  will  be  equal  to  1.     Hence  in 

any  such  case  the  rationalization  will  lead  directly  to  the  integration.     But 

if  — — 1  is  a  negative  integer  in  the  former,  or  — \-  —  -\-l  is  a 

n  n  q 

positive  integer  in  the  latter,  the  exponent  of  ^  —  b  will  be  negative,  and 

the  rationalization  will  not  generally  lead  to  the  integration  ;  and  in  fact  it 

is  not  usTially  expedient  to  rationalize  in  such  cases. 


126  THE  INTEGRAI.  CALGDLUS. 

184:,  Cob. — Every  differential  of  the  form  dy==  Ax'"(a+bx)''dx  can 
he  rationalized  and  integrated  when  either  m  or  p  is  a  positive  integer. 

Dem. — If  p  is  a  positive  integer,  a-j-bx  can  be  expanded  into  a  series  of  a  finite 

number  of  terms,  whicli  multiplied  by  x'"dx  will  give  a  series  of  monomials  ;  and 

an  algebraic  monomial  can  always  be  integrated  by  {IGS  or  1S6). 

.           ,      .            ,       ,        .^.       m4-l       ,       m-|-l       , 
If  p  is  fractional  or  negative  and  m  integral  and  positive, 1  =  — 1, 

will  be  a  positive  integer  or  0,  and  Prop.  %  will  effect  a  rationalization  which  will 
lead  directly  to  the  integral. 

1 
Ex.  1.  Integrate  dy  =  x^{2  +  ^x'^^dx. 

Sug's.— Since  m  =  5,   and  n  =  2,  ^^^-^t i  —  2,  a  positive  integer,   and 

n 

Prop.  2,  will  lead  to  the  integration. 

To  rationalize,  put      2  +  3x-  =  z^j  (1) 

i 
whence  (2  +  3a;2)-=z,  (2) 

Differentiating  (1)       xdx  =  \zdz,  (3) 

~3 — /'  ^^^ 

Multiplying  (2),  (3),  and  (4)  together 

dy  =  x-{2  +  Sx^fdx  =  -M^'  —  '^)^^^dz  =  -^{z^dz  —  4:Z^dz  +  Az^dz). 

,-.  y  =  ijfiz^dz  —  4^'idz  +  4z^dz)  =  5^,/^  —  -|-  -f-  -|- j  +  C;  or,  restoring  the 

value  of  .,   2,  =  -A- 1  '^^^  -  i^^4^^  +  ^^4^*  [   +  a      [ThU 

result  may  be  expanded  and  reduced,  if  desired.  ] 

3. 
Ex.  2.  Integrate  dy  =  x^{a  +  bx^)^dx. 

Ex.  3.  Integrate  dy  =  x^{a  —  x^)    ^dx. 

2/  =  —  J(a  —  x')^2a  +  x')  +  G. 

1 
Ex.  4.  Integrate  dy  ===  x^{a  -^  x)^dx, 

y  =  ^2^(a  +  x)^{5x-^  —  Aax+  f  a^)  +  0. 

dx 
Ex.  5.  Integrate  dy  = 


1' 

^4(1  _^  a;2)2 


"- J  =  J  x-*{l  +  a-)  ~(ir.     Here  m  =  :—  4,  n  =  2,  and 

p  =  -X;  whence  !!!L±i+^4-l  =  ^i-±i  _  ^  +  X  =  ^  1,  and  Prop.  3, 
will  lead  to  the  integratioii. 


BATIONALIZATION.  127 

l^utting  1  +  fl?2  =5  z«a;«  (1) ;    we  have  x^  = (2)  ;    x  == (3) ; 

sr-<  =  (z^  —  iy  (4);  l+.r2  =  l+-^i-^=-^-  (5)  ;  (l+a;2)""*  =  z-i(z2_l)^  (6); 
and  differentiating  (3),  dx  =  —  (z2  —  1)  ^z  cZ2  (7)^ 


Multiplying  together  (4),  (6),  and  (7),  there  results 

3x^ 


y  =  /^-4(1  +x2)  *da;  =  — /(z2  —  l)(2z  =  z  —  iz3 _[_  c  =  (2a;3  — l)(14-.r2)   _^  ^^ 


Ex.  6.  Integrate  cZi/  = 


3 


.  (1  +  ^.Ni^^    .  ^ 


ax 


Ex.  7.  Integrate  dy  =  a(l  +  ^2)    ^dx.  y  ==  — -^ — —  4-  G. 

(1  +  x"")^ 
Ex.  a  Integrate  dy  =  a7-*(l  --=  ^x'^)~'^dx. 

1   _L  4j;2  1 


IRRATIONAL    FRACTIONS. 

18 S,  JPvop,  1, —  When  a  fraction  contains  none  but  monomial 
surds,  it  can  be  rationalized  by  substituting  a  new  variable  toith  an  expo- 
nent which  is  a  common  multiple  of  all  the  denominators  of  the  fractional 
indices  in  the  given  expression. 


Dem. — The  general  form  of  such  a  fraction  is 

m  p 

ax"  -\-  hx'^  -\-  etc. 


dx. 


r  t 

a'x'  -\-  b'x''  -f-  etc. 

m  p  r 

In  this  put  X  ==  z"9'*"»  ®*'=-  ;  whence  a;"  =  z'"^su,  etc.^  ^q  -_.  2«iwttj  etc-,  x'  =  i^vm,  etc.^ 

jpit  __  ^nqrt,  etc.^  ^nd  dx  =  (ngsM,  etc.)  z"?*";  etc.  —^dz.     These  values  substituted  in  the 
given  fraction,  evidently  render  it  rational. 

180,  Cor. — This  method  is  equally  applicable  when  the  fraction  in- 

m 

volves  no  surd  except  one  of  the  form  (a  +  bx)  °,  6y  treating  a  +  bx  as 
the  variable. 


187*  I*rop,  2,— When  a  fraction  contains  no  surd  but  one  of  the 
form  v/a  -j-  bx  ±  x%  U  can  be  rationalized  by  putting  v/a+bx-j-x*  =  z— x 


128  '.  THE   INTEGllAL   CALCULUS. 

when  x'  is  -f- ;  and  when  x^  is  — ,  s/n  +  bx  —  x^  ==  \/(x —  ^){p  —  x) 
=  (x  —  a)z,  in  which  a  and  fi  are  the  roots  of  the  equation 
a  +  bx  —  x^  =  0. 


Dem. — 1st.    When  x^  is  +.     Putting  >/a  -{-  bx  -\-  x'  =  z  —  x,  a  -{-  hx  -{-  x"^  = 
■2zx-\-X' ;  whence  x  = 
2-'  —  a        z'^  -{-  hz  -\-  a. 


z2  —  2zx-\-X' ;  whence  x  =  - — —7,  dx  = ! -^ dz,  and  Va  A-bx-\-  x^=^ 

'Az-\-b  {'Az  -\-  by-  '  ' 


-.     Hence  as  x,  V'a  -i-  bx  -{-  x\  and  djx  are  expressed  in 


rational  terms  of  z,  the  transformed  fraction  \viii  be  rational. 


.    2nd.    When  x-  is  — .     Assuming  \/a-]-bx  —  x^  =  V  {x  —  a)(/i  — x)  =  {x—  a)z, 

and  squaring  we  have  (x  —  «)(/?  —  x)  =  (x  —  ay^z%  or  /3—  x=  (x  —  a)z^ ;  whence 

az-i -\- /5    .        2(a  —  /3)zdz        ^      . — — 1^22  +  ^  ) 

x=.^-^,dx==-^—,^nd   ^a  +  bx-x^==   ]_il_«^.= 

''  .^  _■   ■■  -.     Hence  as  x,  \/a-\-bx  —  x^,  and  dx  are  expressed  in  rational  terms  of  z, 

the  transformed  fraction  will  be  rational,     q.  b.  d. 

i  2 

Ex.  1.  Integrate  dy  =  dx. 

5x^ 

A  8 

Stjg's.— Put  x==z^;  whence  d^/  =  Y^'  dz  —  ^^z^  dz.     y  =  -^^x    —  |x'  +  C 

1 

Sx^dx 
Ex.  2.  Integrate  dy  == — -' 


2072  _  a;! 


{5.        2  1  •  \ 

/pb         07^        ^X  1  1  if 

-g- + -2  + -3- +  4a;^  + 16^"^— 32  log  (2 — o;*)  [  +  C. 


Ex.  3.  Integrate  dy  = 


Sitg's.— Putting  1  -\-  x  =  z^,dy  =  -^  —  ^  _J    ;  whence  3/  ==  2  tan-^l  +"  «) 
-f  C. 

Ex.  4.  Integrate  dy  = 


(1  +  4a7)^ 

,=^r(iiif)!_3(i+4.)4 — 3    ^__i_^n^^_ 

^       "^  (l  +  4r)^     3(1  +  40?)^-' 

Ex.  5.  Integrate  dy  =  —  2/  =  log    /—^= h  C: 

.rv/l  4-07  \/l  +  ^  +  1 

SuG.  — jEr's  4  and  5  can  be  performed  hy  {184)  or  {181).     In  fact  these  methods 
are  essentially  identical  when  there  is  but  one  surd  of  the  form  (a  -}-  ftx)". 


Ex.  6.  Integrate  dy  = 


EATIONALIZATION.  •  129 

dx 


xVl  -{-  X  -{-  x^ 


Solution.  — Put    \/l  -j-  ic  +  .r-  =  z  —  x;    whence  z  =  x  -\-  \/l  -|-  cc  -j-  x\ 

X  = -,  dx  = ■ — ---^ ,  and  VI  -\-  x  -{-  x^  =  —■ — — -^ — .    Substitut- 

ing  these  values 

/dx  /'2(z24-z  +  l)        2z  +  l  2z+l    ,  /*  2dz 


log g^  +  C: 

2  +  a;  +  2^1  +  a;  4-  ;»^ 

CfcJ7 

Ex.  7.  Integrate  dy  = 


V  X-  —  X  —  1 

y  =  log  [c{2x  —  1  4-  ^V'x-^  —  x  —  l)]. 


__     ^    _  ,         ,     _         dx\/2x  4-  ^2 
Ex.  8.  Integrate  dy=  — 


x^ 


Sug's. — ^Putting  \/2aHh^  =  z  —  ic,  there  results,  in  the  nsual  way  dy  = 

z2_|-4z+4  ,             z^-^z       .    4(z-f  l)dz          (Zz       ,    4dz  .      ,    ,  -.n 

=  — ■ • — dz  = — ^^ — ■ =  — -—  A .      .'.  y  =  log (24-1)  — 


-  4-C=  log  (cc  4- 1  4-v/2.'c  4-  ^=^)-  — r— =rr=  4-  0. 
2  -^4-  \/2^'  4-  x^ 


dx 
Ex.  9.  Integrate  dy  = 


\/  2  —  :;;  —  x"' 


Solution. — Put  \/2  —  x  —  x"  =  \/\x  -j-  2)(1  —  x)  =  (x  4-  2)z  ;  whence  x  = 

1  — 2z'2    ,  6zdz  ,     , 3z  r  dx 

2-2  +  1  (z24.l;2'  z24_l  ^      J    ^2  —  x—a;i 


-/r 


4^  =  — 2tan-iz4.(7=— 2tan-i      ?_-^  4- C. 

dx 


Ex.  10.  Integrate  dy 


V^l  4-  a;  —  x^ 


Suo's. — From  1  -{-  x  —  a;^  =  0,  we  learn  that  the  factors  are  x  —  (^-.-f-  i>/&)  an4 
(h  —  i\/5) —  X.  As  these  roots  are  so  cumbrous  it  will  be  economy  to  take  x  —  a  and 
^  —  Of  as  the  factors,  as  in  the  general  demonstration.     The  differential  in  terms  of 

«  is  dy  ==  —  :j-¥^.     .-.  y  =  — 2tan-i2  4-(7==2eot-V  ii:-i^?^Z^4-C'. 


130  THE  INTEGRAL  CALCULUS. 

SUCTION   IV. 

Integration  by  Parts. 

188.  The   Formula  for   Integration   by  !Part8  is 

Ju  dv  =  uv  —  fv  du. 

This  formula  is  deduced  directly  from  the  differential  of  a  product.     Thus  d{uv) 
=  vdu  -\-  udv  ;  whence  uv  =  jv  du  -f-  ju  dv,  and  ju  dv  =  uv  —  fv  du, 

FORMULA]    OF   REDUCTION.  jfiL,    50,   <Sr,    AND   ^. 

189,  JProh, — To  produce  a  formula  for  reducing  the  exponent  of 
X  without  the  parenthesis  by  the  exponent  of  x  within,  in  the  form 
dy  =  x"'(a  +  bx")''dx  ;  i.  e.  to  make  the  integration  of  this  form  depend 
ujjon  the  form  J'x"'"'"(a  +  bx")Pdx. 

SoiiUTioN. — The  solution  of  this  problem  is  effected  by  applying  the  formula  for 

integration  by  parts  to  the  form  dy  =  x'^ia  -j-  bx'*)vdx.     To  make  the  application, 

put  dv  =  {a  •}-  bx^'jPx'^'^dx,  whence  u  =  x'^-'^+K 

ra  _|_  ?)x")''+' 
Integrating  the  former  and  differentiating  the  latter,  we  have  v  =  — p j — - — , 

and  du  =  (in  —  n  -|-  l)x"'—"dx. 

Substituting  in  the  formula  fu  dv  =  uv  —  J^v  du,  we  have  y  =  fx"'{a  -f  hx'*)Pdx 

= i^ — -L 1— -    x^-Ha  4-  hx^y+^dx.*    (1). 

Now    rx"'-"(a  +  hx")p+^dx  =  fx'^-^a  -j-  hx")Pdx  X   («  +  ^*")  = 

afx'"-^{a  +  6x")Pdx  -f  bfx'^a  -f  hz'')Pdx. 
Introducing  this  value  of  fx'^'-^ia  -f-  bx")P+^dx  in  (1) ,  we  have 

.r"*— "+i(a  4-  bx'^)P+^       m  —  n  -^  1     r         ,      .   ,     v    , 
If       J      ^     ^        '  nb{p  -}- 1)  nb{p  -f- 1)    -^ 

r~ —  I  a;»»(«  +  bx'')Pdx. 

Transposing  the  last  term,  we  have    j  1  -j ■ — tt pr  [  J  a;'"(«  +  &a;«)J'da:  = 

x-^^^^(^a4-bx^^r^^-(^^-nArr)aS^--i<^^^^^^T^  ^,,  („^+,h+1)  fx^ia^^^^ydx 

rib{p  4-1)  *' 

_   a;»'-"+i(a  -|-  bx")p+^  —  a{m  —  ?i  4-  l)Ja'"'~"(^  4-  bx")Pdx  ^        a     ^    i\ 

'  b 

r     ,     ,,    .   ,        ix^-^+Ua4-bx^)P+^  —  n(m  —  n-^l)jx'^-^(a-\-bx")pdx 
y=     a;"*  a  4-  &x»)Pdx  = ^^ — ' — r- j r^-r^ •     («£i» ). 


*  Tliis  operation  diminishes  m  by  n,  but  it  also  increases  p  by  1.  The  latter  may  be  a  disad- 
vantage. It  will  evidently  be  more  likely  to  prove  advantageous  if  we  can  diminisb  or  increaie 
either  m  or  j>  without  affecting  the  other. 


INTEGRATION   BY   PARTS.  131 

1.00.  JProh, —  To  produce  a  formula  for  increasing  the  exponent  of 
X  without  the  parenthesis  by  the  exponent  of  x  within,  in  the  form 
dy=:x'"(a+  bx")Mx;  i.  e.  to  make  the  integration  of  this  form  depend 
upon  the  form  /x'"+''(a  +  bx°)Pdx. 

Solution. — Clearing  (jA^)  of  fractions,  transposing  the  first  member  and  the  last 
term  of  the  second  member,  and  dividing  by  a(m  —  n  -f-  1)>  we  have 

j  a;'^"(a  4-  bx')Pdx  = —^ J—^ LJT i — . 

Putting  m  —  n  =  m',  whence  m  =  m'  -\-  n,  this  becomes 

r     ,,     ,,     ,   ,         a;"»'+i(o4-6ar»y+i — i5>(np+m'+n-|-l)  ra;"»'+"(a4-!/iC")PcZa: 

y  =  j  x«  '(a  -f  hT^ydx  = ^^ — ■ ■■ —^-^ ■ ■ — -^ -— L i • 

aim  -j-  1) 

or,  dropping  the  accents, 

y=jx^{a-{-bx^)pdx= — — 7-..        ' ^-^- — —'  (2S.) 


191,  J^rob. — To  produce  a  formula  for  diminishing  the  exponent 
of  the  parenthesis  by  1,  in  the  form  dy  =  x"'(a  +  bx")Pdx  ;  i.  e.  to  make 
the  integration  <f  this  form,  depend  upon  the  form  J*x™(a  +  bx'')P~'dx. 

Solution. — ^We  may  write 
y  =  fx'^-{a  4-  hx'')pdx  =  fx"*(a  -f-  &x»)p-»da  X  (a  -f  &a;")  = 

afx"'{a  4-  hx^)p-^dx  -f-  bfx'^+^a  -f-  bx'')p-Hx.  (1). 
By  applying  formula  (J^)  to  the  last  integral,  it  becomes 

r     ,   ,      ,   ,     ,       ,         x^'+^a  +  &x«)p  —  aim  +  1)  fiK'^ra  +  bx'*)p-^dx 
j  x'«+"(a  +  5x")p-icZa;  =  ■ ^ ■ — -^ ^^ — — . 

Substituting  this  in  (1)  it  becomes  y  =  fx"'{a-j-bx'')Pdx  =  afx'^{a-^hx")P—^dx  -f. 

a;'«+i(a  _}-  6x")p  —  a(m  +  l)j"x«(a  +  bx'')p-^dx  ^ 

np  +  m  4-  1 
or,  uniting  terms, 

r  ,        x'»+i(a  4-  bx")p  +  anp  fx'^ia  -\-  bx^)p—'^dx 


102,  J?VOh, — To  produce  a  formula  for  increasing  the  exponent  of 
the  parenthesis  by  1,  in  the  form  dy  =  x'"(a  +  bx'')Pdx  ;  i.  e.  to  make  the 
integration  of  this  form  depend  upon  J'x"(a  +  bx^)P"''^dx. 

Solution. — Transposing  and  reducing  (<^)  in  the  same  manner  as  we  did  (^) 
in  producing  (§3),  we  have 

r     ,      .   ,     .       ,         x-^+\a  -f  bx'*)P  —  (np  -\-  m-i~l)  fx'^ia  4-  bx'')Pdx 

•^  '       —anp 

Patting  j>  —  t  xszt>\  ^bonce  o  ssr  j?'  4.  l,  this  becomes 


132  THE  INTEGEAL  CALCULUS. 

*       -^      ^     '  —  ani^y  -f- 1). 

or,  dropping  tlie  accents, 

^       -^      ^    '       '^  — a7i(p  + 1) 

ScH. — Binomial  differentials  of  this  form,  or  such  as  may  be  readily  re- 
duced to  it,  are  of  such  frequent  occurrence,  and  the  formulae  S^,  H^,  <gp, 
and  5|>,  called  Formulce  of  Reduction,  are  so  frequently  efficient  in  reducing 
them  to  known  forms,  that  these /ormwZce  should  be  carefully  memorized. 

Ex.  1.  Integrate  dy  == j-. 

/x^dx  f  ~\ 
— ^    =  J  a;2(a2  —  a;2)    ^x,  a  form  which  corresponds 

to  fx^{a  -j-  6x'')Pda;,  by  considering  m  =  2,  n  =  2,  a  =  a^,  b  =  —  1,  and  p  =  —  ^. 
We  now  observe  that  if  the  exponent  of  x  outside  the  parenthesis,  or  in  the  nu- 
merator in  the  given  form,  were  0,  so  that  x^  =  1,  the  integral  would  be  a  cir- 
cular function.     Now  formula  {^)  will  so  reduce  this  exponent ;  hence  we  apply 

/'v"dx  c  ~  5 

— : ^  =  J  a;2(a2  —  x'^)    dx 
(a2  —  a;2)^ 

a;2-2+i(a2  _  x^)~^'^^  —  052(2  —  2  +  l\r.'3;2-2(a2  _  x2)~^da; 


-l[2(-^)-i-2-hl] 
i 
—  2     "    '    2 


a;(a2  — £c2)2       a2  ^  -i 

=-^a;(a2  _  a;2)^  +       / ^ 

•/    (a2  — jc2) 
i      .      o^    .       ^   .    « 


Ex.  2.  Integrate  c??/  = '- r 


Ex.  3.  Integrate  dy 


(1  _  a;2)2 
3/  =  —  ia;2(l  —  5r2)i  _  1(1  _  ^,)i  _|_  a 


(2  +  a;2)f 


y  =  a;«(2  +  a;^)  "^  +  4(2  +  a?')  ^  +  C'=     "^^  "^  ^  .  +  C. 

(2  4-  ^'^r 


INTEGRATION  BY  PARTS.  133 

Ex.  4.  Integrate  dy  = '■ — ~. 

(1  — a;«)2 

SuG. — Apply  jfikj  twice  in  succession,  and  we  have 

/I       ,1-4       ,    l-2.4\   , 

Ex.  5.  Integrate  dy  = 


(1_^2)2 


/I         15         135\  /- 1-3.5.     , 


dx 
Ex.  6.  Integrate  <f t/  =  - 


x\l  —  a?2)2 

1x^  4-1  i 

SUG.-Apply  50  twice,    y  = ^(1  -  ««)'  +  G 

Ex.  7.  Integrate  cZy  = 


si 


Solution,     y  =  T =  /cc-3(—  a^  -f  a;2)~  dx.    Now  by  ii 


dec 

increasing 

a;3(ic2  —  a2)'^ 

the  exponent  of  x  without  the  parenthesis  by  that  within  the  form  becomes  known. 
Hence  we  apply  (^).  In  this  case  m,  =  —  3,  n  =  2,  a  =  —  a%  6  =  1,  and 
p  =  —  ^.     Hence  substituting  in  the  formula 

7  =J  ar-3(— a2  -j_  a;2)  *  (to 

_  a;-3+i(— ag-fa;2)~^^^  —  [2(—  ^)  —  3+24-l]/ar-3+2(— g2-|-a;2)~^(fa; 
~  —  a2(—  3  +  1^ 


2a2  / 


2a2  '   2a2  /    ,   „        „,i- 

(a2  — a2)2'        1  a;  ,    ^ 

-2^ir-  +  2T3«^^-^a  +  ^- 


Ex.  8.  Integrate  dy  =  (a^  —  a:2)2:r2<fa?. 
Sug's. — Applying  j^,  we  have 

Applying  <Sf  to  J  (c^  — -  a;«)*dx,  we  have 


134  THE  INTEGKAL  CALCULUS. 

h  ^ 

/(a2  _  x-^fdx  =    ^    +  2/(0^  -  a:«)  *da;  =  -L_— i-  +  |  sjn-i  ^+  C. 

3.  1. 

T/  «         oxi  oj             a;(a2  — a;2)*    ,   a5x(a2  — £c2)^   .   «-»  .       a; 
•••  y  =  J  (a2  —  a2)*x2dx  =  —         ^     '^   +  ^ ^  +  g  sm-i-  +  a 

Ex.  9.  Integrate  dy  =  {1  —  x^)^dx. 

SuG.— Apply  <g  twice,     p  =  \x(l  —  x^)*  -j-  |a;(l  —  x^y  +  f  sin-i  x -{-  C, 


Ex.  10.  Integrate  dy  = 


(a2  -f  a;2)»* 


SuQ.— Apply  m    V  —  ;; 1 f(a^  4-  x8)-ida;  =  -^ \- 


2a^J  a-^ 


^  ^  +„4-,tan-.?+a 


+  a;2       2a2(a2  +  x^)   '    2a3  a 


Ex.  11.  Integrate  dy  =  -. 

^  ^        (1  +  a;2)3 


^  3         a?  3^       , 


^       4(l  +  x2)2  '   8(1  4-  ^=^)       8 

ScH. — Tihese/oy'mulce  often  fail,  in  consequence  of  reducing  the  expression 
to  00,  by  making  the  denominator  0  ;  or  by  making  the  expression  indeter- 
minate.    Thus  it  would  seem  at  first  glance  that  formula  S  would  reduce 
.1 

;  but  it  will  be  found  to  fail.    Nevertheless  the  formulm 

X 

are  of  great  practical  value ;  and  that,  not  only  in  such  examples  as  the 
above,  which  they  reduce  directly  to  the  elementary  forms,  but  in  many  more 
complicated  cases  where  they  reduce  the  given  expression  to  a  form  which 
can  be  integrated  by  methods  yet  to  be  given. 


LOGARITHMIC   DIFFERENTIALS. 

193.  JProb.—To  irUegrate  the  form  dy  =  X  •  log"  xdx,  in  which  X 
is  an  algebraic  function  of  x. 

Solution.— Put  Xdx  —  dv,  whence  log»x  =  u,  and  substitute  in  fudv  = 

uv—fvdu.    Thus  y  =  fX'  log«xdc  =:log"x./j:da;— /rnlog"-Jx/(X  dx)---! 

If  now  fXdx  is  a  known  form,  we  have  made  the  integration  to  depend  upon  a 

form  in  which  the  exponent  of  logx  is  diminished.     Thus,  if   fXdz  =  X,  the 

X' 
form  of  the  expression  to  be  integrated  becomes  —  log"-i  x  dx.     To  this  the  form- 

X 

via.  may  be  applied  again,  and  n  diminished  still  farther  if  the  algebraic  function 
-—dx  can  be  integrated. 


Ex.  1.  Integrate  dy 


INTEGRATION  BY  PARTS,  135 

lo^  X  dx 


SuG. — ^Put  -- — ; ■  =:  dv  ;  whence  logjc  r=z  u,  v  =  —  :; — ; — ,  and  du  ==  —. 

(1    -|-  X)^  °  1  -|-  X  X 

losr  X  dx  log  X     .      /*      dx 


/"log  X  dx  log  ^-     .      f      d^ 


dx 
Separating   —  -  into  partial  fractions  {174:),  and  integrating,  we  have 

Xyl.   -\-  X) 

FinaUy,  y=-i^+loga;-log(l+a;)  +  a=^loga;-log(14-x)  +  a 

Ex.  2.  Integrate  dy  =  logxdx,  y  =  ^(loga;  —  1)  +  C 

Ex.  3.  Integrate  dy  ==  a;2log2  j;  dx. 

y  =  ij73(log«ar  —  flog  a;  +  |.)  +  G. 

dx 
Ex.  4.  Integrate  dy  ==  — ; . 

^  ^        a:  log' a; 

Solution.— Put  dv  =  —  ;  whence  u  =  log-2  x,v  =  log  x,  and  dw  =  —  2  log-^  a;--. . 

X  •  iC 

/*    dx  1 

Substituting  in  the  formula  for  integration  by  parts  y  =    j    ,   '^    ,  =  ,         + 

/*    d^  /*    dx  1  1 

2  I  -^ .     Transposing  the  last  term  —  /  — — —  =  :j ,  or  y  =  —  t— — 

J  jclogsa;  t'       Q  J  ajlogsa;      logo;        "^  logo; 

Ex.  5.  Integrate  dy  =  — r-dx. 


a 


X' 


2 

y  — i(log'^  4-  41oga?  +  8)  +  a 

x^ 

_          ,   .         ,      ,          x\os,xdx 
Ex.  6.  Integrate  dy  = ^^^ -. 

1 
y==  (a2  4-a;2)2loga7  +  «log— ^-^ — - — {a^ -\- x^y -\-  G. 


EXPONENTIAL    DIFFERENTIALS. 

19 4:,  T^rob, — To  integrate  dy  =  x'^a^'^'dx,  when  n  is  a  positive  in- 
teger, 

Sol.— Put  dv  =  a'^'dx  ;  whence  u  =  x",  v  = a'^,  du  =  nx'^-^dx.     Sub- 

'  c  log  a 

stituting  in   the  formula    fudv  =  uv  —  fvdu,   we  have    y  =  fx"a"dx  = 

aci/gii  _  _ fx^—^a'^dx.     Applying  the  formula  to  Jo;"— 'cr'da?,  the  inte- 

c  log  a  c  log  a-^ 


136  THE  INTEGRAL  CALCULUS. 

gration  is  made  to  depend  upon  the  form  ^J'a;"-2««da;.     Thus,  the  exponent  of  x 

can  be  finally  reduced  to  0,  and  the  integration  made  to  depend  upon  the  form 

r                                1 
A  I  a^'dx,  which  =  — a"  A-  C. 

•^  c  log  a 

Ex.  1.  Integrate  dy  =  x^ef'^dx. 

Solution. — Put  dv  =  e'^dx  ;  whence  m  =  x^,  v  ==  -&^,  and  du  =  Zsd^dx. 

/x^             3  r 
x^ef^dx  =  —  e"* J  x^^dx,       (repeating  the  process) 


x^ 

—  i 

a  a- 

a3  3a;2      .  6 


= —ga* e,<^-\ — \  x&'^^dx      "         '*        *• 

a  a2  a'^J 

a;3           3x2           6x           6  /-      , 
=  _e«2 e*^  _j Qdx e'"dx 

x^  3x2        ,    6x  6         ,    ^ 

a  a2  a^  a* 

/x3       3a;2   ,    6x       6\    ,    ^ 

=  e«^| )  4-  CI 

\a        a"       a^       a*/ 

Ex.  2.  Integrate  ^2/  =  ^^(fdx. 

a^    (  3.272  6^  6     )         ^ 

y  ==  n 1  X'  —  :; h  1 1 r   +  ^• 

log  a  (.  log  a      log2  a       log^  a ) 

Ex.  3.  Integrate  dy  =  e'x^dx. 

y  =  e'ix*  —  Ax^  +  12a;2  _  24ar  +  24)  +  C. 

x"dx 
Ex.  4.  Integrate  dy  =  — --. 

SuG.— Put  dv  =  e-'dx,  as  before.     y  =  —  e-*(x2  +  2x  +  2)  -f-  (Z 


SPECIAL   FORMS   OF   EXPONENTIALS. 

e"'  —  1 
Ex.  5.  Integrate  dy  =  —^ — —dx. 

dx  =  /  ; dx  =  log  [c(e*  +  e-^)],  as  (e=^  —  e-^)(Zi3 

q2x  _^1  f     ga:   _|_  g— z 

=  d(e^  +  e-^). 

Ex.  6.  Integrate  dy  =  e^'e'dx. 
SuG.— Put  e'  =  z,  whence  dy  =  e'dz,  and  yz=^  +  0=e^'  +  0. 

Ex.  7.  Integrate  cZ^/  =  ,.    ,   '     e^'dx. 


SoLUTioN.  •  Put  1  -|-  X  =  z  ;  whence  x  =  z  —  1 ,  and  dx  =  du 

/»  1  _L  x"-      ,  / 

Substituting,      y  =    I  - — ■ e'^dx  =   / 


1  _L  x"-    ,        rz-'  4-  2  —  2z     , , 


INTEGKATION  BY   PABTS.  137 

l^  +  ^f-^-f"^} 

— ,  by  putting  dv  =  2-2^2, 

re'dz            e'   ,     re'dz 
and  u  =  e",  we  nave   I  -—  = f-   I   — . 

Substituting  this  value, 

^  =  i*'  -  T  +  y  -  -  y  -J  =  V'  -  t)  +  ^=  ^V  -;)+<^= 


Ex.  8.  Inteerate  dy  =  -—— — -V-  V  =  "^ — r~~  +  ^• 


TRIGONOMETRICAL    DIFFERENTIALS, 

19  S.  JProb, —  To  integrate  the  forms  dy  =  sin^xdx,  dy  =  cos'^xdx, 
dy  =  sin""  x  cos"  xdx. 

Solution.  —To  integrate  dy  =  sin"«a;dx,  put  sinx  =  z  ;  whence  cosx  =  (1  — z^)  , 

dx= — —  =  (1  —  z2)  2^2  ;   and  we  obtain  dy  =  z'^CL  —  z^)     dz,  which  may  be 
cos  a; 

rationalized  by  {181,  182),  or  reduced  by  one  or  more  of  formulae  <^,    ^, 

<Q,  and  W- 

In  like  manner  putting  cos x  =  z,  dy  =  cos" xdx  becomes  dy  =  —  z"(l  —  z^)  *dz, 
and  can  be  disposed  of  as  before. 

m 

Again,  putting  cosx  =  z  ;  whence  sin"»a;  :^  (1  — z^)^y  cos^a;  =  z",  and  da;  = 

dz  — 1- 
=  —  (1  —  z^)  '^dz,  we  have 


(1  —  z^Y 

m—X 

dy  =  sin"  x  cos«  a;  da?  =  —  z"(l  —  z*)  *  d« ; 
or  we  may  put  sin  x==z,  and  have 

n— 1 

dy  =  sin'»a;cos"a;da;  =  z"(l  — z^)  ^  dz. 
Either  of  these  forms  may  be  treated  as  the  first. 

196.  ScH. — ^It  will  be  seen  that  this  process  will  always  effect  the  integra- 
tion when  m  and  n  are  either  positive  or  negative  integers,  and  frequently 
when  they  are  fractions.  Thus,  when  m  and  n  are  positive  even  integers, 
successive  applications  of    0^    will  reduce  the  final  integral  to  the  form 

=fc:   r(l  — z^)     dz  =  ±:  f i- —  =  sin-^z,  or  cos-^2;  =  x ;  and,  when  posi- 

•^  J    {\-z-^)^ 


138  THE  INTEGRAL  CALCULUS. 

tive  odd  integers,  the  final  form  will  be  i  fz{l  —  2fi)  dz  =  =F  (1  —  ifiy  =s 
=P  cosic,  or  =f  sin  a;. 

When  m  and  n  are  negative  integers,  formula  ^  will  reduce  the  first  two 
cases. 

The  third  case  may  require  any  or  all  of  the  iour/ormulce,  but  can  always 
be  integrated  when  m  and  n  are  integers. 

In  many  cases  it  will  not  require  the  appUcation  of  the  formulos,  as  will 
be  seen  in  the  following  examples. 

Ex.  1.  Integrate  dy  =  sin3  x  dx. 

SoiiUTioN. — Putting  since  =  z,  and  applying  ^,^e  have 

r                      r                 -i .         22(1  —  z2^^  _  2  rz(l  —  z^)^dz 
y  —  j  sm^xdx  =  J  z\l  —  z2)  ^dz  = —^ — ■ 

=    _  iz2(l    _   z2)^    4_    |Jz(l    _   z2)~^ck 
=   —iz\l    —   Z2)^    _    1(1    _   Z2)2    _^    a 

=  —  i  sin2  X  cos  cc  —  |  cos  a  -f-  C. 

Ex.  2.  Integrate  dy  ==  sin^  a;  dx. 

eosx,  .  ,   .      .        ^  ^ 

2/  = -— (sin^a?  +  f  smar)  -f-  f  a:  -f-  t7. 

Ex.  3.  Integrate  dy  =  sin^  x  dx. 

2/  = — ^(sin^a:  +  fsin^ar  +  f )  +  C^. 

5 

Ex.  4.  Integrate  <^2/  =  sin«  x  dx. 

cos  ^ 
y  = ^— (sin^a;  +  l-sinaa;  4-  ^sina;)  +  -^x  -f  C. 

D 


Ex.  5.  Integrate  dyz=cos^xdx. 

y  =  i  (sina;  cos x  -\- x)  +  (7  =  |-  (|- sin 2x^  -{-  x)  ■}-  C, 

Ex.  6.  Integrate  dy  =  cossj^^or. 

2/  =  ^  sin  ^cos2  X  +  %smx  -\-  (7  =^  ^^  sin  3^7  +  |  sin  a;  +  C.f 

Ex.  7.  Integrate  ^i/  =  cos^  x  dx. 

y  =  ^ig.  sin  4a;  -f  i  sin  2^7  -f  f  j7  +  (7. 

Ex.  8.  Integrate  dy  =  cos^  j?  dx. 

sinar  N       ^      ,/sin5a7     5 sin 3.57  \,  ^ 


*  Trigonometry  (56)  sin  a;  cos  x  =  ^  sin  '2x. 

t  To  effect  the  reduction    substitute  1  -  sin2  x  for  cos2  x ;    and  then  for  sins  x  subBtitut© 
l(Z  sinx  —  sin  3a;).     (See  Trigonometry,  page  28,  Ex.  12.) 


INTEGRATION   BY   PARTS.  139 

Ex.  9.  Integrate  dy  =  cos^ x  ^m!^ x dx. 

SuG  s. — Putting  sm  x  =  z,  there  results  y  =  — - — f  i 1 —  1  +  (7. 

_  ^,.                                ,                        Gosf'x/.        cos2.r    ,    cos'<.'c\    ,    ^, 
Putting  cos  X  =  z,  we  nave  y  = ; — (  t ^ 1 = —  j  -f-  ^  • 

Query. — What  is  the  relation  between  Cand  C  ? 

Ex.  10.  Integrate  dy  ==  sin^  x  cos^  x  dx. 

.        /,        sin^aTx        ^ 
y  =  sm'^^^l —j  4-  a 

SuG. — If  one  factor  has  ah  even  and  the  other  kh  odd  exponent,  it  will  b6  found 
expedient  to  put  the  function  which  has  the  even  exponent  =  z. 

Ex.  11.   Integrate  dy  =  sin^^r  cos^xdx. 
Solution. — Put  sin  a?  =  z,  and  apply  4^. 

Z5(l   _   22)2  1  _i 

Now  apply  gfik^  to  the  last  integral. 

1 

Z\\  —  22)    ^^dz  =  _i-__i-  4-  I J  Z2(l  _  z-2)    ^dz 

i 

3z(l  — z2)^    ,     3    r  .  .-i", 

s=s  ««        ^ —A (1 — z2)  ^dz. 

•■•  2/  =  fUl  -^')^  +  ^(1  -  2^)^}  -  -^^— {^^'  +  W-  Tksin-iz  +(7  = 


Bin"  .T 


n  'V  ^  .  COS  ^  I  "^ 

—  {cos^jK  -f-  acosjc}   —  -— -— {sin^ic  -f-f  sina;}  -f-rir*  +  ^• 

64: 


Ex.  12.  Integrate  ^v  = —dx. 


3 


Sug's.— Putting  sin  a;  =  z,  we  have  dy  =  z^{l  —  z^)  ^dz^ 

Applying  ^  twice  y  =  o (^iii'*  a  +  ^  sin2  x  —  8)  -^  C. 

o  cos  X 

dx 
Ex.  13.  Integrate  dy  =  — 


sm^x 

Sug's  . — Put  sin  x  =  z,  and  we  have  dy  =  z—={l  —  z^)  ^dz. 
Apply  H^  twice  and  we  have 
1 

)^ 

4:Z^  "  2z' 


^  i 


(1—22)2  (1— Z2)^       .       ,r  .  ^-^, 

y  =  —       ^^^  -  —  I  .  —2^-  +  f  J  2-Hl  —  2')     (^2,  or  restoring  x, 

4    Vsin-^a;       2siu-x/         j    sin  a;' 

=::  _  ^I(  J-  +  --^  )  +  5»  log  tan  (^x)  -f   a     (See  ^59,  1. ) 
4    Vsin-'x       2sin-^ar/    '    •     *        v^  /    i 


140  THE  INTEGRAL  CALCULUS. 

dx 


Ex.  14.  Integrate  dy 


cos®^ 


y  = 


sin  x/  1  4,8 


C       d  COS3  /r  i5  COS  X/ 


5    Vcos^a;     Scos^j;       3  cos  a; 


StJG. — Applying  5S  three  times  the  last  term  reduces  to  0,  and  we  have  the 
result  without  integrating. 

dx 
Ex.  15.  Integrate  dy  = 


sm*  X  COS2  X 


Solution. — Putting  cos  a;  =  z,  whence  sin— ^x  =  (1 — «*)~',  cos— ^a;  =  z— 2,  and 
iz,  we  have 


-4 
dac  =  —  (1  —  z^)     dz,  we  have 


/dv  r  r  -9 

-. "■ =   j  cos-2a;sin— *a;dx  =  —    f  z-2(l  —  z^)     dz. 
SUl'*  X  cos2  a?         -^  -^ 

Applying  ^, =  -  J i -f  4/(1  -  z^Y'dz  \ 

*  -  Z(l  -  Z2)2  J 

"  2&, =  ^-4]   f _    +    §/(l_,2)-^dJ 

Z(l  —  Z2)^  '  3(1  —  22)=^  ' 

"  .        "  again.  -  = ^ ^1-^  _  f  ] 1— j-  -  0  I  +  C. 

Z(l  —  Z2)^  .3(1  —  22)^  ^  (1  _  22)2  ) 

T.    ^    .  14  cos  X       8  cos  a;   .    „ 

Eestormg  x, = : — -: \-  C. 

cosicsm^a;       Jsm'^a;       3  sin  a; 

Anotheb  Solution. — When  the  exponents  are  even,  an  elegant  solution  is  ob- 
tained by  means  of  a  special  expedient,  as  follows  : 

Introducing  the  factor  sin^  x  -f-  cos2  x,  which  being  equal  to  1  does  not  change 
the  value  of  the  differential,  we  have 

div /•(sin2.r  4- cos2a;)da;  r        dx  \      C   ^     

sin4  X  cos2  X         I         sin4  x  cos2  x  /  sin2  x  cos*  x         I  sin-*  x 

(sin2  X  -f-  cos2  x)dx    ,      /*   dx  /*   dx       .       r  dx       ,       r  dx 


sin^a; 


/(sm2  X  -f-  cos2  x)dx    ,     f   dx     r   dx  r  dx  r 

sin2a;cos2a;  '     /  sin-'a;         /  cosmic  I  sia^x  I  & 

,  ,  cos^r    1        ,       2    "I    ,    ^ 

=  tan  X  —  cot.r 4-  C. 

3    Lsin^a;       sin.'rj 

[It  will  afford  the  student  a  good  exercise  in  trigonometrical  reduction,  to  trans- 
form the  former  expression  into  the  latter.  ] 

dx 
Ex.  16.  Integrate  dy  = 


sin  X  cos3  X 

y  == f.  log  tan  a?  -f  O. 

Ex.  17.  Integrate  dy  =  dx  =  tan-*  x  dx. 

cos-^^ 

SuG. — ^Put  tancc  =  z,  whence  dx  = = .     .-.  y  =  (  —dx  = 

sec2a;       1  +  z2  '      J  cos*  a; 

dz 

iz3  —  z  4-  tan->  z  +  C  = 


/tan^xdx  -y^pi:,  =  fzHz  -  fdz  +fjj^^, 
itan'x  —  tanar-fx-f  C. 


INTEGRATION   BY   PARTS.  14:X 

Ex.  18.  integrate  dy  =  tan^  xdx, 

y  =  ^iQji*x  —  ^tan2j7  +  -J- log  sec  a;  +  (7. 


197.    JProb, — To     integrate    the    forms    dy  =  x"sinxdx,    and 
dy  =  x"  cos  X  dx. 

Solution. — To  integrate  dy  =  a^'sinxdx,  put  dv  =  Binxdx,  whence  u  =  x"*, 

u  =  — cosic,   and  du  =  nX^-'^dx.      .-.  y  =  —  jc^cosa;  -f  wj^x" -' cos .r dr.     By 

repeating  the  process  the  integration  will  finally  depend  upon  the  form  ^  fcosxdx* 

or  A'  jsmxdx. 

In  like  manner  y  =  fx"cosxdx  =  a;"  sin  a;  —  nf  x''-^  Bin  xdx,  and  the  final 
forms  are  the  same  as  before. 

Ex.  1.  Integrate  dy  ==  x^  cos  xdx. 

y  =  x^sinx  -{-  Sx^  cos  x  —  6^  sin  ^  —  6  cos  :r  +  C. 

Ex.  2.  IntegTate  dy  ==  x'^  sin  x  dx. 
y  ==  —  ^^coso;  +  4a;3sina7  +  Vlx'^cosx  —  24^  sin  j;  —  24cos^  +  G. 


198.  Prop, —  When  m  and  n  are  integers,  the  form  dy  == 
sin""  X  cos"  X  dx  may  be  integrated  in  simple  terms  of  the  sines  and  cosines 
of  the  multiple  arcs. 

DEM.~The  form  sin^a;  cos"a;  may  be  expressed  in  simple  terms  of  multiple  arcs 
by  the  use  of  the  following  formulae  : 

(1)*  sin  £c sin?/  =  \  cos  {x  —  y)  —^  cos  {x  -f  y), 

(2)  cos  X  cos  y  =  ^  cos  {x  —  y)  -\-^  cos  {x  -\-  y), 

(3)  sin  X  cos  y  =  i  sin  {x  +  2/)  +  2  sin  {x  —  y). 

The  truth  of  this  statement  and  the  manner  of  applying  the  formulae,  may  be 
seen  most  readily  from  the  solution  of  a  few  examples. 

Ex.  1.  Integrate  in  simple  terms  of  the  sines  or  cosines  of  multiple 
arcs  dy  =  sin^  x  cos^  x  dx. 

Solution,     sin^  x  cos^  x  =  sin  x  (sin  a;  cos  xY 

z=  sin  x[^  sin  2.'r]2     [From  (3),  making  y  =  ;v.  ] 
=  \  sin  ir[sin2  2x'] 
-^'r=  L  sinic[^  cos  0  —  ^  cos4£c]      [From  (1),   making  x  =3 
y  =  2a;.] 
=  ^  sin  xW  —  |-  cos  4a?] 
=  \  sin  X  —  \,  sin  x  cos  Ax 
=  \  sin  X  —  \{^  sin  5a:  —  |-  sin  3a;]      [From  (3),  making 

X  =  X,  and  y  =  dx.] 
=  "li  sin  x  —  -i\-  sin  5x  -f-  iV  sin  3x. 
Hence  y  =  J  sin^  x  cos2  xdx  =  i  fsin  xdx  —  J-jJ'sin  5xdx-\-  -^gj^ sin  Sx  dx  = 
^  -^  eos.r  —  -4^8  cos  3x  -f  -g'o  cos5x  +  C. 

*  These  formulas  are  essentially  those  of  Art.  S9,  Plane  Trigonometry.    To  put  the  formultt 
of  that  article  into  this  form  simply  change  *  into  ^{x  -\-y)  and  y  into  j^{x  —  y). 


142  THE  INTEGRAL  CALCULUS. 

Ex.  2.  Integrate  in  simple  terms  of  the  sines  or  cosines  of  mnltiple 
arcs,  dy  =  sin^  a;cos3  x  dx. 

Sug's.  siii3  X  cos^  a;  =  s  sin^  2x  =  \  sin  2a;  sin2  2a;  =  §  sin  2a;(^  —  ^  cos  4,x)  = 
-jig  sin  2a;  —  -j^^  sin  2a;  cos  4a;  =  ^^g  sin  2a;  —  -/g(^sin6a;  —  ^  sin  2a;)  =  i^-^  sin  2a;  — 
■^  sin  6a;  +  ^  sin  2a;  =  ^^  sin  2a!  —  -^^  sin  6a;. 

.  • .  y  —  f  sin^  X  cos^  xdx  =  y\  y  sin  2a;  d!a;  •—  ^*^  y  sin  6a;  <Za;  =  —  ^  cos  2a;  + 
Ys^^  cos  6a;  +  (7. 

Ex.  3.  Integrate  in  simple  terms  of  the  sines  or  cosines  of  multiple 
arcs  dy  ==  sin"  x  dx. 

Sug's.  sin^  x  =  (sin2a;)3  =  g(l  —  cos  2x)^  =  §  —  |  cos  2x  -f  §  cos^  2a;  —  g  cos^  2a; 
=  5  —  I  cos  2a;  +  1(5  +  i  cos  4a;)  —  gcos2.i;(5  -j-  ^  cos  4a;)  =  -^  —  -i^gcos2a;  -j- 
-j^^  cos  4a;  —  -^g  cos  2a;  cos  4a;  =  -i%  —  i^g  cos  2a;  -)-  i^^^cos  4x  —  -^^{^  cos  2a;  +  a  cos  6a-) 
=  -j%  —  4jf  cos  2x  -{-  i\  cos  4a;  —  aV  cos  6a;. 

.-.  y  =:  aVC—  d  sin 6a;  +  f  sin 4a;  —  J/ sin 2a;  -]-  10a;)  +  ^'  [The  student  should 
be  careful  to  observe  that  the  three  formulae  given  under  the  proposition  are  suf- 
ficient to  effect  the  required  reductions.] 

Ex.  4.  Integrate  as  above  dy  =  cos^  x  dx. 

1/sin  Sx       ^    .      \       ^ 
y  =  4V— 3 h  3  sin  ^J  -f  ^. 


CIRCULAR    DIFFERENTIALS. 

100*    JPvob, — To     integrate     the    forms     dy    =  f (x)  sin""*  x  dx, 
dy  =  f(x)  cos""^  X  dx,  dy  =  f  (x)  tan~^x  dx,  etc. 

Method  of  Solution. — Put  f{x)dx  =  dv,  and  sin— 'a;,  cos— ^a;,  or  tan-' x,  ais  the 
case  may  be,  =  u,  and  substitute  in  the  formula  for  integration  by  parts. 

Ex.  1.  Integrate  dy  =  x-  sin~^  x  dx. 

dx 


SoiiUTiON. — Putting  a;2da;  =  dv,  whence  sin— ^  x  =  u,  v  =  ix^,  and  du  = 


Vl  —  x-i 

/*     't;"(Z.^ 
we  have  y  =  ix^  sin— 1  x  —  if  — ^' — ^ — .     Applying  formula  ^  to  the  last  inte- 

J    \/l—x'-      

gral  we  have  y  =  ix^  sin-i  x  +  I  {x-  +  2)  \/l  —  x'^  +  G. 

Ex.  2.  Integrate  dy  =  \/l  —  x^  cos~*  x  dx. 

Solution. — Putting  (1  —  x-Ydx  =  dv  ;  whence  u  =  cos— ^x,  v  s=  1x0.  —  x-)    + 

.  r    (i^  *    .  •,      s-^    ,  -,  7  ^'^ 

f  /  — z=z=z  =  '2^(1  —  ^  )"  —  -|  COS— 1 X,  and  aw  = . 

J    s/l  —  xi  Vl  —  a;=2 

Substituting  in  the  formula  for  integration  by  parts, 

■■'  Ty  applying  *^ 


INTEGBATION  BY  PABTS.  143 

y  =  ^[a;(l  — a;2)^  —  cos-i  a;]  cos-i a  — /^ [.r(l  —  x^)^  —  cos-»a;](—    -    ^^  — ) 

V  v/l    />.2/ 


^1  —  X2' 

4-\)xdx4-l  Jcos— 'a/ i 

-f-  ioj'-^  -{-  -^  fcos— '  a;  d(cos— '  cc) 


=  ^[a;(l  —  £k2)^]  cos-1  .r  —  i(cos-ia;)2  +  ix'^  +  C. 

Ex.  3.  Integrate  dy  =  -^- —r-^dx, 

^  ^  1  +  ^•'^ 


?/  =  tan~'^  (a:  —  i  tan""^  x)  —  log  Vl  -\-  x*  +  ^• 


200*  JPvob, — To  integrate  the  forms  dy  =  e"'' sin"  x  dx,  and 
dy  =  e"''  cos"  x  dx. 

Method  of  Solution. — Put  e«*da;  =  dv:   whence  sin" a:  =  w,  v  r=  -e"*,  and 

a 

dtt  =  n  sin"— ^  X  cos  a;  dr. 

1        .              n  r        . 
.' .  11  =  —&"'  sm"  X I  e"^  sm"-!  x  cos  a;  c7a:. 

Applying  the   formula   for  integration   by  parts   again,   putting   dy  ==  C'dr, 

u  =  sin"— 'a; cos X,    v  =  — e"^,   du  =  (n  —  1)  sin"— 2 .r  cos^ a;  dx  —  sin".rcZ;r,    and 

a 

we  have  '  .• .  y  =    j  e"^ sin" x dx  =  — e"* sin" x  — 

n  \1        .       ,  n  —  1  r       .  ,.lr       .,) 

-  i  -e"'"  sin«— 1  x  cos  x e"=^  sm"— 2  cos2  xdx  -i —  I  e«^  Bin"x  dx  Y  . 

a  i  a  a     -^  a-^  ) 

Transposing  the  last  term,  uniting  it  with  the  first,  and  dividing  by  —^ — ,  we 
have 

<*           .               n           .       ^              ,  nin — 1)  r       ^  o      > 

y  =  — ; — e^^sin^x ; — e^'^sm"— ^xcosx-J I  e^^'sm"— -a;cos^  xdx. 

If  now  the  last  term  be  transposed  and  united  with  the  first  member  and  we 
divide  by  the  coefficient,  we  shall  have  made  the  integration  to  depend  upon  a 
form  in  which  n  is  diminished  by  2. 

By  successive  applications  of  the  same  process,  the  integration  may  be  made  to 

depend  upon  the  form  Aje'^^dx  when  n  is  even,  or  A'fe'^^  sin  a;dx  when  n  is  odd. 
The  former  is  an  elementary  form  ;  and  the  formula  lor  integration  by  parts  being 
applied  to  the  latter,  it  becomes  an  integral  without  further  process,  since  the  co- 
efficient of  the  unintegrated  term  contains  a  factor  n  —  1,  as  will  be  seen  above. 
By  a  process  altogether  similar  the  form  d'?/  =  e<^  cos^  x  dx,  can  be  integrated. 

Ex.  1.  Integrate  dij  ==  e""  cos  xdx. 

N  Solution. — Put  dv  =  C'^x  :  whence  u  =  cos  a?,  v  ==  -e"'',  and  du  ==  —  sinxdai 

a 


144  THE  INTEGRAL  CALCULUS. 

y  =  je"'  COS  xdx  =  -&"=  cos  x  -4 —  fe<^  sin  x  dx 
*^  a  a-' 

1  ,   1      •  1  r 

=  -e""  ees  x  A e«^  sm  a; I  e<^  cos  x  dx. 

a  a-  a^^ 

.'.  y  = -——-e'^ cos X  4-  — — -— — e*^ sin  a;  +  C. 
(a  cos  a;  -}-  sin  x)  -^  C. 


a^  +  1 


Ex.  2.  Integrate  dy  =  ef"  sin^  a;  c?a;. 

2/  =  tV^(s^^^  ^  +  3cos3  x  4-  3  sin  x  —  6  cos  a;)  -f  C. 

Ex.  3.  Integrate  <?2/  =  ^~^  sin  ^ar  cZa;. 

a  sin  ^j:  +,  Z;  cos  hx 


dx 
20  !•  I^voh, — To  integrate  the  form  dy 


(a  +  bcosx)*" 

/*  du  /*(«  4- 6  cosxldx  /•  cte 

Solution,     y  =  I  --; r  =  /  — r r-r;  =  «  /  z — r-i: ttt  + 

/   (a  4- 6  cos .x)"        /   ^a  4- 6cosx)»+i        ^/   (a4-6cosx)"+i    ' 

/*        cos  "^  fZ^ 

h  I ^-^^ .     Applying  the  formula  for  intesrration  by  parts  to  the  last 

J   (a  4-  &  cos  .T/'+i         t-f  J     &  s  J  r 


integral  by  putting  cos  xdx  =  dv,  we  have 


J 


(a4-^cosiC;"+i       (a4-^cosic)"+i 


/•      h^in^xdx        sin  a:  .    ^.  /*<&  —  6  cos2-x)da; 

''     W   (a  4-  6  cos  xy'+'  ~  (a  4-  6cosa;j"+i  J   («  4-  ^  cos  iCj«+2' 

Substituting  this  value  in  the  preceding  we  have 

/*        dx  r  dx  6  sin  a;  /*(b^ — b'-cos^at^dx 

(aH-6cosx)»~^/  (a4-6cosa;)''+'  "^  (a4-6cosa;)«+»  /   («4-6cosa;)"+a 

?>  sin  a;  ,         /•  da; 

— -\-  ci  t — 

(a  4"  ^  cos  .x/'+ '  /    (^a  4-  6cosa;)"+' 

rh"'  —  a"'  A-  'la'. a  4-  ^cos.r'i  —  [a  -\-  6cos.r)2*  ,  6  sin  a* 

(n  4-  1)  /    ^ ^ ■ ■ dx  = ; h 

^      '      J  (a  4-  6.cosx;"+'-  (a  4- 6cosic/'+i 

/»  /7t  '^  dor  /*  d'V 

+  (n  +  l)f       ,f ^. 

^  ^    ^      /   (a4-6cosa;)» 

Transposing  and  uniting  similar  integrals, 

r  dx  bsinjc -I_1^ /* <?g 

(n4-l)(&2  — a2)  /  ^^_^^cosa;)"+2  ~  (a4-6cosa;)«+i  ~  ^V  ^+   ^  (a4-6cosa;)«+» 

-+-  n  I ^. 

'      /   («  4-  6cosa;)» 

Dividing  by  (n  4-  1)(&-  —  «-).  and  writing  7i  —  2  for  n,  we  obtain 

*  By  adding  and  sxibtracting  2a2  -f  2a6  cos  «,  62  —  b2  cos2 1  =  &2  -j-  2ai  -f  2a&cos  »  —  2a2  — 
2  lb  C03  X  —  b?  C082  2;  =  b2  —  a2  -(-  2a(a  4-  b  cos  t)  —  (a  -i-  b  cos  x}^. 


'^h 


dx 


INTEGRATION  BY  PARTS. 

6  sin  a;  a(2n  —  3) 


145 


dx 


-f-6cosx)»       (w— 1)1,6- — a-)(a-{-6cosxj"-i       (n — 1)(6"^ — a^)l  (a+6cosa;)"-i 

n  —  2  /•  dx 


+ 


t^)J  (( 


(n  —  l){b'^  —  a'^)J  {a  -}-b  cos  iCj"-=*' 

By  means  of  repeated  applications  of  this  formula,  or  the  process  by  which 
it  was  produced,  the    integration    may  be  made    to    depend    upon    the    form 

1  a  -\-o  cos  X 

(It  X  x 

To  integrate  dv  == "■ ,  we  remember  that  cos  x  =  cos^  -  —  sin=^  -,  and 

a  -)-  6  cos  X  4  2 

cos2  -  -j-  sin2  jr  =  1.     Hence 


.=/: 


dx 


a  -{-  b  cos  X 


A 


dx 


cos2  -  -\-  sin2 


i)  +  < 


/ 


dx 


X 


{a  -{-  b)  cos2  5  +  (a  —  &)  sins  -    = 


/^       sec2  -  dx 
«y        .     a  4-  6 


(a  — i»^ 


(«-&) 


-d(tan|) 


X        a  -}- 


X  .   ,x 

cos2  -  —  sin2  - 

dx 


cos2 


(«  +  &)  +  («  —  ^) 


COS2- 


2ci!(^tan  |  j 


X  +  -^tan.|      «.-.,* 


tan 


,    a  —  b  ^     „x        a  -\-  b 
a  -{-  b  2 


-,  J i^LZ^^^tan^  j.  +  C,  when  a>6. 


y 


When  a  <1  &,  we  have 
dec 


/     a+6  cosa;         /    , 


sec2    da; 


flj 


(&+a)  —  (&  — a)tan2- 


ib  4-  a)' 


i(tan|) 


(6+a)^— (&  — «rtan| 


/. 

V    (£ 


2cZi 


(tan  I) 


(6-i-a)  —  (5  — a)tan2- 

_l-^(tan|) 

(6  +  af 


(64-ar+(&  — a)Han- 
(6— ard^tan|j 
(52_a2f/  (&-fa)^  — (6  — a)^tan^        {b'^—a^?  I   (&  +  «)^  +  (5  —  a)* tan ^ 


(&  — a)^dAan^j 


T— ^  + 


(1*2  —  a2) 


log 


1  J  £C 

(6+a)^ -f-  (6  —  a)-  tan  -  ' 


+  a 


■146  THE  INTEGEAL  CALCULUS. 

ScH. — The  four  preceding  sections  comprise  the  greater  part  of  what  is 
known  concerning  abstract  methods  of  passing  from  the  differential  to  the 
exact  integral  function  of  a  single  variable ;  but  there  are  many  other 
methods  of  great  practical  importance  for  determining  the  approximace 
value  of  the  integral  of  a  differential  which  cannot  be  integrated  by  tne^e 
methods.     One  of  the  most  useful  and  simple  is  given  in  the  next  section. 


■♦4» 


SUCTION   V, 
Integration  by  Infinite  Series. 

202,  It  often  occurs  that  a  differential  can  be  expanded  into  an 
infinite  series,  the  terms  of  which  can  be  integrated  separately  If 
the  result  is  a  conyerging  series,  the  value  of  the  integral  may  be  de- 
termined with  sufficient  accuracy  for  practical  purposes  by  summing 
a  finite  number  of  terms.  It  may  also  sometimes  happen  that  the 
law  of  the  series  is  such  that  its  exact  sum  may  be  found,  although 
the  series  itself  is  infinite. 

This  method  is  not  only  a  last  resort  when  the  methods  ol  exact 
integration  fail,  but  it  is  sometimes  serviceable' by  being  more  smiplt 
than  they,  even  M^hen  they  are  applicable  ;  moreover,  it  affords  a 
method  of  developing  such  a  function. 

11 
Ex.  1.  Integi-ate  dy  =  a;^{l  —  x^)^dx. 

Solution. — Expanding  (1  —  a;^)^  by  the  binomial  or  Maclaurin's  theorein,  w:^ 

have 

(1  _  x^f  =  1  _  ia;2  _  ia;4  _  ^^^s  _  _A_a.8  _  etc. 

^  \  1  pi  p  5.  p  a.  r  ^?-  r  ■^^' 

. •.  ?/  =  j  x^{l—x'^ydx=J  x'dx—^J  X  dx—^J  x^dx--^j  x^  dx—^j  x''-  ax— etc 

3.  2.  JLL  15.  ill 

=  fcc^  —  V  —  ^4^     —  Tioae  ^  —  T2^««     —  etc.  +  G. 
This  series  is  converging  for  a;  <C  1  and  more  rapidly  converging  as  x  is  less. 


Ex.  2.  Integi'ate  dy 


dx 


(1  4-^4)2 


,  ^5       1  -  3  x9       1  •  3  -  5  a7'3   ,      ^        •  ^ 
2/  =  .:_^-  +  _^-_^^^gj3+etc+a 


Ex.  3.  Integrate  dy 


2 

(x  —  l)'^dx 
x^ 


6-1         .1         2.-4  8     -^         .      ,   ^ 

y^-x^^  4^^  +  __;^  ^  _  —X    ^  —  etc.  +  C, 


SUCCESSIVE  INTEGRATION.  147 

dx 
Ex.  4.  Integrate  dy  =  • ^  in  an  infinite  series,  and  thus  obtain  a 

development  ot  y  =  tan~'^. 

/dx         ,       ,  ^      ^  x^      x'^      x"^      x^  _ 

\  dx 

Ex.  5.  Integrate  dy  =  —  in  an  infinite  series  and  thus  obtain 

Vl  —  x^ 

a  development  of  y  =  sin~^^. 

/dx  .     ,         ^  .  073      3a75       5^7 

7r^^  =  sm-:r+C=^+^  +  j^+j^-  +  etc.  +  (7. 

Ex.  6.  Integrate  dy  = in  an  infinite  series  and  thus  obtain  a 

development  of  i/  =  log  (14-^). 

/*  dx  ,      /-.         s       xw  072      a?3       a;^        .  _ 


■♦♦» 


SUCTION'   VI. 
Successive  Integration. 

203,  JPvob, — To  integrate  a  second,  third,  or  nth  differential  func- 
tion of  a  single  equicrescent  variable. 

Solution. — Since  dx  is  constant  we  may  write  jd^y  ==  ff{x)dx'^  =  dxj'f{x)dx, 
and  integrate /(jc)dx  as  before,  thus  reducing  the  degree  of  the  element  (differential) 

by  unity.      Putting  yf{x)dx  =/,  (a;)  +  (?i  j  we  'bavP!  yd-y  =  yf{x)dx'^  ==  dxff{x)dx 

z=f^{x)dx  -\-  Cydx.     But  fd-y  =  dy  ;  hence  dy  =/,  ix)dx  -j-  Cj dx. 

Integrating  again  we  have  2/  =/2(a;)  +  CiX  -\-  Cz- 

In  like   manner  from  d^y  =f{x)dx'^,  we  have  J'd^j/  =  d^y  =  dx^Cf{x)dx  = 
dx'±f,{x)  +  CJ  =/,(a;)c?x2  +  G,dx^. 

Integrating  again, 
j'd^y  =  dy  —  dx  ffi{x)dx  -\-  dxfC^dx  =  dxlfiix)  -f-  C^x  -\-  C^]  =/2(cc)c?a;  + 

Cixdx  -{■  C-idx. 

Integrating  a  third  time,  we  have 

fdy  =  y  =  fMx)dx  +  G,fxdx+  C^  fdx  =Mx)  +  hC,x^  +  C^x  +  C,. 

From  these  processes  we  deduce 

*  By  preceding  methods.  t  ^Hiis  notation  sigxdfied  the  nth  integtaL 


148  THE  INTEGRAL  CALCULUS. 

Ex.  1.  Integrate  d^y  =  6a  dx\ 

Solution,     fd^y  =  d^y  =  fSa  dx^  =  6a  dx^fdx  =  6ax  dx^  -f-  C'l  *dx*. 
Again,    jd^y  =  dy  =  Jl6axdx^  -\-  Cidx^}  =  Badxjxdx    -|-  Cidx^dx  = 

Zax-dx  -j-  C^xdx  -\-  C-idx. 
Finally,  y  =  ax^  -\-  iC^x"^  +  C^x  -\-  Cg. 

Ex.  2.  Integrate  <i^</  =  cos  x  dx*. 

PBocEss.f    d^y  =  sin xdx^ -]-  Cidx^, 

d^y  =  —  cosxdx^  +  C^xdx^  -\-  Czdx'^, 

dy  =  —  Binxdx  -f-  iCiX^dx  +  Czxdx  -\-  C^dx, 

y  =  cos  a;  +  ^C^x^  +  iCaCC^  +  C-^x  +  C4. 

ScH.  1. — It  will  be  observed  that  whatever  value  we  assign  the  constants, 
we  get  the  original  differential  by  differentiating  the  integral  as  many  times 
as  we  integrated.  The  reason  for  this,  if  not  seen  at  once,  will  appear  upon 
performing  the  operation. 

d^y 
Ex.  3.  Given  -7-^-  =  0,  to  find  the  integral  y. 

Solution. — Since  the  third  differential  coefficient  is  the  differential  of  the 
second    differential    coefficient,    divided    by    the    differential    of    the  variable, 


d{ 

d^y  \dx^/       r^       ^V        (.   d^y  .  -u  ,     ,     -a  •     +i,- 

t.  e.  -TT—  =  — r; ,    when   -- —  =  0,  —^  must  be  a  constant.     Hence  m  this  ex- 

dx^  dx  dx^  dx'-i 

d^v 
ample,  y^  =  ^1  >  or  d^y  =  Cydx^.     From  this  we  obtain  y  =  IC^x^  -{-  C^x  -\-  C3. 

Ex.  4.  Given  -v—  ==  —  to  find  the  inteeral  y, 
dx^       x^  ^       ^ 

y  =  log^  +  ^C^x^  +  C^x  -}-  G^, 

Ex.  5.  Integrate  d^y  =  sin  x  cos*  x  dx^. 

Solution. — Putsina;  =  v;  whence  dv  =  cosxdx,  cos^xdx'^  =  dv%  and  d^y  =' 
sin  X  cos2  x  dx^  =  v  dv^.  From  d-y  =  v  dv-,  we  obtain  as  before  y  =  ^v^  +  6'i i?  -j-  Cj . 
.  • .  2/  =  i sin3 a;  -j-  ^1  sin  x  -\-  Cg. 

Ex.  6.  Integrate  d^y  ==  cos  x  sin^  x  dxK 

y  =  ^  COS3  X  -}-  C^  cos  X  +  Cg. 

ScH.  2. — In  order  to  integrate  successively  d"y  =/{x)dx'',  according  to  the 
foregoing  process,  it  is  necessary  that  we  be  able  to  integTate  exactly /(.r)r/r, 
fi[x)dx,  f^{x)dx,  etc.  It  is  evident  that  this  will  be  frequently  impossible. 
A  method  of  approximation  which  is  often  serviceable  in  such  cases,  is 
readily  obtained  by  means  of  Maclaurin's  Formula. 

*  We  might  write  GaCi  as  the  coefficient  of  this  term  ;  but  as  Cl  represents  any  constant,  it  is 
unnecessary  to  retain  the  6a. 

t  This  is  simply  a  convenient  form  in  which  to  write  the  operation  ;  the  student  should  under- 
stand the  process  and  be  able  to  explain  it,  as  in  the  preceding  solution. 


SUCCESSIVE   INTEGRATION.  149 

204,  JProp. —  The  nth  integral  of  f(x)dx"  may  be  developed  into  a 
series  by  the  following  formula : 

^ ^x";;;^^ ,   ^^^^^^         x"  ^  fdf(x)-  -"+^ 


l-2.3---(n-l)+'^('')]l-2-3---n  +  L   dx  Jl  •  2  -  3  -  -  -  (n  +  1) 

rd2f(x)-i  x"+-  rd3f(x)-|  x"+' 

"^L   dx-^  Jl.2-3---(n+2)"^L~dx3  Jl-2-3---(n  +  3)+'^*'^' 

Dem.  —1st.  Developing  y  =  fy{x)dx"  by  Maclaurin's  formula,  we  have 
y  =  fy{x)dx^  =.  ify{x)dx'^^  +  [/""y(;r)dT«-^]*|  +  [/"■y(x)dx»-^]^l. 


•3---(n  — 1)' 

2nd.  Comparing  this  development  with  the  development  of  fy{x)dx''  as  made 
in  the  solution  Akt.  203,  remembering  that  .'C  =  0  in  the  bracketed  factors  of  the 
present  series,  and  that  these  factors  are  therefore  constant,  we  see  that  the  present 
series  is  that  of  Abt.  203  reversed,  extended  and  generalized  ;  that  [fyix)dx"] 
is  Cn,   [j  "~y(a5)dx"— 1]  is    Cn—i,  etc.,  and  that  the  former  series  begins  with  the 

term  [/(cc)]r— •.—,•; of  the  latter.     Hence  using  C,„  (7„_i,  etc.  for  these  constant 

i-A-o <n 

factors  the  above  development  becomes 


(^^^r^TT^-TZ — -.-.  +[/(x)]nr^ r  + 


fdAx)-!  x«+i 


1.2.3---(n  — 1)    '    '-•'^  '•'1.2-3---n    '     L  dx  Jl.2.3 --- (n  +  1) 
rd^fix)-\  x-+^  p^(.r)-|  a;"+3 

■^  L  dx'-^  Jl.2.3---  (n  +  2)  "^  L  dx^  Jl.2.3--- (n  +  3)  "^  ^^'''     ^-  ^'  ^' 

ScH. — This  formula  is  readily  remembered  and  applied  by  noticing  the  law 

of  the  first  part  of  this  series  (that  containing  the  constants  C„,  Gn—i,  etc. )  ; 

and  then  observing  that  the  second  part  of  the  series  {that  from  and  includ- 

x"' 
ing  [/(a;)]- — —^ },  is  the  development  oi/{x)  by  Maclaurin's  Formula, 

each  term  being  multiplied  by  x",  and  the   successive  terms  divided  by 

1  -  2  -  -  -  7z,  2  -  3  -  -  -  (n  +  1),  3  -  4  -  -  -  (w  +  2)  respectively. 

^       _       ,  ^.  ,  r*     d^* 

Ex. 


Develop    y  =    C^d'^y  =   I  - 


-/I 


x^ 


1  -^ 

SoiiUTioN.— Here  we  have  n  =  4:,  &ndf{x)  =  — r=rzr  =  (1  —  x^) 

\/l—x^ 

_i 

Developing  (1  —  x^)   ^  by  Maclaurin's  Formula  we  have 

*  To  differentiate  an  integral  is  to  depress  its  order,  as  will  be  seen   from  the  natiire  of  the 
processes. 


150  THE  INTEGRAL  CALCULUS. 

(1  -  x^)-^  =  1  +  ia;=  +  h.^x^  +  ^TTtI'^  +'  ^^' 
Hsnce  by  the  above  scholium,  we  obtain 

1-3x8  ,  1.3.5a;io 

+  n    ..    ^    ^    rr    o    n    .A  +.   etc. 


1.2. 4.5-6. 7-8  ^  l-2-4:.6.7-8.9.10 


-♦♦» 


SECTION    YIL 

Definite  Integration  and  the  Constants  of  Integration. 

205,  Def. — An  Indefinite  Integral  is  one  in  which  the 
constant  or  constants  of  integration  remain  undetermined,  and  which 
has  not  been  satisfied  by  any  particular  value  of  the  variable. 

LCiL. — All  the  integrals  hitherto  produced  are  indefinite  integrals. 

206.  Def. — A  Corrected  Integral  is  one  in  which  the  value 
of  the  constant  (or  constants)  of  integration  has  been  determined 
and  substituted  for  the  general  symbol  G. 

207 •  Def. — An  integral  is  said  to  be  taken  between  Limits ^ 
when  the  indefinite  integral  has  been  satisfied  for  two  different  values 
of  the  variable,  and  the  difference  between  these  results  taken. 

208.  r>EF. — A  Definite  Integral  is  an  integral  taken  be- 
tween limits. 

A  definite  integral  and  the  limits  between  which  it  is  taken  is  sym- 
bolized thus  :  y  =    I  f{x)dx.     This  signifies  that  the  indefinite  inte- 

gral  ot  f{x)dx  is  to  be  obtained,  and  first  satisfied  by  substituting  h 
for  X,  then  by  substituting  a  for  x ;  and  that,  finally,  the  latter  is  to  be 
subtracted  from  the  former,  x  =  a  and  x  ==h  are  called  the  limits 
of  the  integral,  the  former  being  called  the  inferior  and  the  latter  the 
superior  limit.* 

Ex.  1.  Find  the  value  oi  y  ==    I     5x^dx. 

Sug's. — The  indefinite  integral  is  y'  =  ^x^  -f-  ^• 

Now  this  is  true  for  all  values  of  x,  hence  for  x  =  3.    For  this  value  of  x,  we  have 

*  It  is  assumed  tliat  x  can  liave  such  values  as  we  assign  it,  and  that  the  function  is  continuous 
between  these  values,  i.  e.  that  it  does  not  become  imaginary  or  infinite  for  any  intermediate 
valua  of  X. 


DEFINITE  INTEGKATION  AND  THE  CONSTANTS  OF  INTEGRATION.  151 

y"   =45  -\-  C.  In  like  manner  for  a;  ==  1 

we  have  y'=    f  -h  ^-  Subtracting  the  latter 

from  the  former,  y"  —  y'"  =  43:j.       . •.  y  =  y"  —  y'"  =    j      5x^dx  =  431. 

Ex.  2.  Find  the  definite  integral  of  dy==nxdx  between  the  limits 
a  and  6.  y  :=    j     nxdx=— — -. 

Ex.  3.  Find  the  value  oi  y  =   j     {x^dx  —  h'^x  dx).     An&.,  — \h*. 


209,  Disposing  of  the  Constant  of  Integration, 

There  are  two  principal  methods  of  disposing  of  the  constant  of  in- 
tegration, G : 

1st.  By  integrating  between  limits,  the  constant  is  eliminated.  This 
IS  illustrated  in  the  preceding  examples. 

2nd.  "When  there  is  anything  in  the  nature  of  the  problem  under 
discussion,  from  which  we  can  know  the  yalue  of  the  function  for 
some  particular  value  of  the  variable,  by  substituting  hese  values  in 
the  indefinite  integral,  the  value  of  the  constant  G  jan  be  found. 
And.  a&  6"  is  a  constant,  if  we  find  its  value  for  any  particular  value 
ot  the  variable,  we  have  it  for  all  values  of  the  variable. 

Ex,    1.     Find    the    corrected     integral    of     the    function    dz    = 
{dx^  +  ^ax  dx^Y,  on  the  hypothesis  that  2  =  0  w^hen  x  =  Q. 

8  ^ 

SuG's.— The  indefinite  integral  is  s  =  x^  (1  +  %axf  +  C.    Now  if  s  =  0  when 

8  8 

a;  =  0,  we  have  O  =  ^^^r-  +  0.    ,'.  G  =  —  ^=—     Substituting  this  value  of  G  in 

8  ^8 

tie  indefinite  integral,  we  have,  as  the  corrected  integral,  z  =  h;=-(1  -{-  %axY  —  x=-. 

27rv  ^ 

Ex  2    What  is  the  value  of  (7,  when  du  =  — ^(^2  _j_  y^ydyyii 

u  =  ^  when  y  =  0  ?  Ans.,  (7  ==  - —  ^Ttp^. 

Ex.  3  Given  du  =  (2'/')2(2r  — •  y)~^dy,  what  is  the  value  of  0,  if 
zi  -=  C  when  y  =zO?     What  if  w  =  0  when  y  =  2r7 

Answers,  Ar^  0. 

Iiiii. — A  differential  is  one  of  the  infinitesimal  elements  of  which  a  quantity  is 
conceived  as  composed.  Thus,  let  A  represent  the  area  of  the  surface  lying  be- 
tween AM  and  AX,  Fig.  35;    PD  being  any  ordinate,  and  a6  the  consecutive 


152 


THE   INTEGRAL   CALCULUS. 


§(2p)y  +  a 


ordinate,  PDdb  may  be  considered  as  representing  an 
element  of  the  area,  or  dA.     This  element  in  the  case 

of  the  parabola  is  found  to  be  \/2pa;  dx ;  hence  dA  = 

\/2px  dx.  This  expression  therefore  represents  any  one 
of  the  infinitesimal  elements  of  which  the  quantity  A 
is  composed. 

The  Indefinite  Integral  is  A 

This  is  indefinite  in  two  respects.  1st,  it  is  true  for  any 
value  of  X  ;  2nd,  it  is  indefinite,  and  in  fact  indetermi- 
nate, as  regards  C,  the  value  of  which  may  be  anything. 

As  KEGABDs  THE  CONSTANT,  if  wc  choose  to  estimate  the  area  from  A,  so  that 
A  =  0  when  x  =  0,  we  have  0  =  0-{-  C.     .  • .   C=  0.     Hence  the  corrected  integral 

1    3. 

is  ^  =  |(2p)^x*.  This  represents  the  area,  estimated  from  A  to  any  value  we  may 
choose  to  give  x.  If  x=  AZ^',  A  =  area  A  P'  D'.  Again,  if  we  choose  to  estimate 
the  area  from  the  focal  ordinate   H  F,   calling  A  =  0,   when  x  =  ^p,  we  have 

|(2p)^(|2>)^  -\-  C,  or  ip^  -^  C.     .  • .   C  =  —  ip"^,  and  the  corrected  integral  is 


Fig.  35. 


0 


A  =  i{2p)^x^  —  ap-'  This  represents  the  area  estimated  from  the  focal  ordinate 
H  F  to  any  value  we  choose  to  give  x.  Thus  if  .r  =  A  D ' ,  this  corrected  integral 
represents  the  area  H  FP'  D'.  Finally,  suppose  we  ask.  Where  must  the  area  be 
conceived  as  beginning  in  order  that    G  =  —  m  ?    To  meet  this  case  we  have 


i.    5. 


0=  |(2p)^a;^  — m,  whence  x 


-<l 


9m  3 
8p~' 


If  therefore  the  area  is  conceived  as  com- 


mencing at  the  ordinate  corresponding  to  jc  = 


w.     Thus  we  perceive 


the  indeterminate  character  of   C.     We  also  observe   the  limits  of  its  possible 
values.     In  this  case  C  may  be  any  negative  quantity,  but  no  positive  quantity. 
Integbation  between  Limits  is  illustrated  by  considering  the  area  as  estimated 

from  some  possible  place  (no  matter  where)  and  extending  1st  to  x  =  a,  whence 

X.   I  13. 

A'  =  H2pra'^  4-  (7 ;  and  2nd  tox  =  h,  whence  A"  =  f  (2p)^6^4-C.  Now  the  dif- 
ference between  A'  and  A"  will  represent  the  area  between  the  ordinates  corres- 
ponding to  X  =  a  and  x  =  h.     Call  this  A'";  and  A'"  =  i{2py{V  —  a'^ ),  if  a  <  h. 

Letting  A D  =  a  and  A D'  =  6,  A'"  =  |(2p)* (6^  —a^)=  area  P D P' D'. 

This  subject  will  have  more  ample  illustration  in  Sections  IX. — XIII.  inclusive, 
which  the  student  is  now  prepared  to  read. 


THE  END. 


w 


'»■ 


The  facility  with  which  the  books  can  be  used  for  classes  of 

all  grades,  and  in  schools  of  the  widest 

diversity  of  purpose. 

Eacli  volmne  in  tlie  series  is  so  constructed  that  it  may  be  used 
with  equal  ease  by  the  youngest  and  least  disciplined  who  should 
be  pursuing  its  theme,  and  by  thos6  who  in  more  mature  years 
and  with  more  ample  preparation  enter  upon  the  study. 


'^O^'J^ 


BOSTON  COLLEGE 


1 


3  9031  01550372  5 


Sheldon  d;  Cofnpany's  Text-Sooks. 

SHAW'S   NEW   SERIES 

ON 

EMLISH  AND  AMEEIOAN  LITERATURE. 


I. 

Shaw'c  New  History  of  English  and  American  JLit- 
ef^ature.  This  book  has  been  prepared  with  the 

greatest  care  by  Prof.  Truman  J.  Backus,  of  Vassar  College, 
using  as  a  basis  Shaw's  Manual,  edited  by  Dr.  William  Smith. 
The  following  are  the  leading  features  of  the  book  ; 

1.  It  haa  been  put  into  the  modern  text-booh  form. 

2.  It  is  printed  in  large,  clear  type. 

3.  Many  parts  of  the  book,  which  were  not  very  clear,  have  been  entirely 
rewritten. 

4.  The  histf 
type,  which  ij 
and  Americai 

5.  It  also! 
remeinber 
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